201 Vector Analysis Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Vector Analysis is one of the most physically rich and mathematically powerful subjects in the engineering mathematics curriculum. It provides the mathematical language for describing quantities that have both magnitude and direction such as forces, velocities, electric fields, heat flux, and fluid flow; for formulating the laws of physics in a precise and coordinate-independent way. For engineering reviewees, a solid command of vector analysis is not optional. The subject appears directly in the board exams for ECE, EE, ME, CE, and ChE, and it provides the foundation for electromagnetism, fluid mechanics, heat transfer, and structural analysis.

This collection of 201 terms covers the complete scope of Vector Analysis as tested in the Philippine Engineering Licensure Examinations. The terms are drawn from four major clusters: vector algebra and geometry, vector differential calculus, vector integral calculus, and the coordinate system representations used in engineering. Each definition is written to be both mathematically precise and practically useful. You will find the formula, the geometric meaning, and the engineering context wherever they matter most for the board exam.

Some terms in this list also appear in Calculus and Advanced Engineering Mathematics, but here they are treated at the full vector analysis level. The gradient is not just a formula, it is a vector pointing in the direction of steepest ascent of a scalar field. The divergence theorem is not just a conversion tool, it is the mathematical expression of conservation laws in integral form. Understanding these terms at this level is what separates reviewees who merely recognize formulas from those who can apply them under pressure.

Work through this list actively and connect each term to a diagram or a physical picture whenever possible. Vector analysis is a subject where geometric intuition and algebraic precision must work together. Reviewees who build both will find that the board exam problems in this area become straightforward once the vocabulary is clear. Use this list as your primary reference for Vector Analysis during your PRC review cycle.

The 201 Vector Analysis Terms and Definitions

1. Acceleration Vector

The second derivative of the position vector with respect to time, a(t) = d²r/dt² = dv/dt. The acceleration vector points in the direction of the net force on a particle by Newton’s second law. In curvilinear motion, the acceleration has two components: the tangential acceleration along the path and the normal (centripetal) acceleration directed toward the center of curvature.

2. Angle Between Two Vectors

The angle θ between vectors A and B, defined by cos θ = (A·B)/(|A||B|). The angle satisfies 0 ≤ θ ≤ π. Two vectors are perpendicular when their dot product is zero and parallel when their cross product is zero. Computing the angle between vectors is a fundamental operation in projection, force decomposition, and the analysis of work done by a force.

3. Angle of Inclination

The angle that a line or vector makes with the positive x-axis in the xy-plane, or more generally with a specified reference direction. For a vector A = Aₓi + Ayj, the angle of inclination α satisfies cos α = Aₓ/|A|. Direction angles and direction cosines extend this concept to three dimensions, giving the angles the vector makes with each of the three coordinate axes.

4. Arc Length

The length of a curve C parametrized by r(t) = x(t)i + y(t)j + z(t)k, computed as the integral of |dr/dt| dt from t = a to t = b. Arc length is a scalar quantity and is the natural parameter for describing curves in a coordinate-independent way. The unit tangent vector is the derivative of the position with respect to arc length, T = dr/ds.

5. Arc Length Parameter

The parameter s measuring the distance along a curve from a fixed reference point, defined by ds = |r'(t)| dt. Parametrizing a curve by arc length gives a unit-speed parametrization where |dr/ds| = 1 at all points. The arc length parametrization is the most natural and geometrically intrinsic way to describe a curve and is used to define the Frenet-Serret frame.

6. Area Element

The scalar or vector quantity representing an infinitesimal piece of surface area used in surface integration. For a surface parametrized by r(u, v), the vector area element is dS = (∂r/∂u × ∂r/∂v) du dv, whose magnitude |dS| = dA is the scalar area element. The direction of dS is the outward normal to the surface. Area elements in cylindrical and spherical coordinates take specific standard forms.

7. Associative Law for Vectors

The property that vector addition satisfies (A + B) + C = A + (B + C). This law holds for all vectors regardless of dimension and ensures that sums of three or more vectors are unambiguous without parentheses. The associative law, together with commutativity, makes the set of all vectors in a space a commutative group under addition.

8. Axial Vector

A vector quantity that changes sign under a reflection of the coordinate system (parity transformation) but not under a rotation. Axial vectors are also called pseudovectors. The angular velocity, magnetic field, and the cross product of two polar vectors are all axial vectors. This distinction matters in physics but is rarely distinguished in standard engineering board exam problems.

9. Basis Vectors

A set of linearly independent vectors that span the vector space, meaning every vector in the space can be expressed as a unique linear combination of the basis vectors. In three-dimensional Cartesian coordinates, the standard basis vectors are i, j, and k, pointing in the positive x, y, and z directions respectively, each with unit length and mutually perpendicular.

10. Bilinear Form

A scalar-valued function of two vectors that is linear in each argument separately. The dot product A·B is the standard bilinear form on R³. In the context of tensors, a bilinear form corresponds to a covariant rank-2 tensor and can be represented by a matrix. The metric tensor of a curved space is the fundamental bilinear form used to compute lengths, angles, and areas.

11. Binormal Vector

The unit vector B = T × N, where T is the unit tangent and N is the principal unit normal to a space curve. The binormal vector is perpendicular to both T and N and completes the Frenet-Serret frame. It points in the direction around which the curve twists. The rate of change of B with respect to arc length is related to the torsion of the curve.

12. Bound Vector

A vector that is associated with a specific point of application, as opposed to a free vector that can be translated anywhere. Forces in statics are bound vectors because their point of application affects the moment they produce. Displacement vectors, on the other hand, are free vectors because only their magnitude and direction matter, not the starting point.

13. Cartesian Coordinate System

A three-dimensional coordinate system in which position is described by three mutually perpendicular axes x, y, and z with a common origin. The unit vectors i, j, k are fixed in direction and form a right-handed orthonormal basis. The Cartesian system is the simplest coordinate system and the one in which the standard formulas for gradient, divergence, and curl take their most familiar forms.

14. Cartesian Form of a Vector

The representation of a vector as A = Aₓi + Ayj + Azk, where Aₓ, Ay, Az are the scalar components along the x, y, and z axes. The Cartesian form is the most common representation in engineering calculations. Addition, dot products, and cross products all have simple formulas in Cartesian form, making it the standard form for direct computation.

15. Cayley-Hamilton Theorem

The theorem that every square matrix satisfies its own characteristic polynomial. If p(λ) = det(A − λI) is the characteristic polynomial of A, then p(A) = 0. The Cayley-Hamilton theorem is used to compute matrix inverses and matrix functions (like the matrix exponential) in terms of lower powers of A without infinite series. It is also used in the derivation of the vector Laplacian identity.

16. Center of Curvature

The center of the osculating circle at a point on a curve. The osculating circle is the center of a circle at any point on a curve which approximates the curve. The center of curvature lies along the principal normal direction at distance 1/κ from the curve, where κ is the curvature. The locus of all centers of curvature as the point moves along the curve is called the evolute of the curve.

17. Centripetal Acceleration

The component of acceleration directed toward the center of curvature of a curved path, given by aₙ = v²/R = κv² = (v·v)κ, where R = 1/κ is the radius of curvature. The centripetal acceleration vector points in the direction of the principal normal N. It is the acceleration required to maintain circular or curved motion and is produced by the net inward force — tension, gravity, or the normal reaction.

18. Centroid via Vector Integration

The position of the centroid (geometric center) of a curve, surface, or solid expressed using vector integrals. For a curve, the centroid position is r̄ = (∫ r ds) / (∫ ds). For a solid region V, r̄ = (∫∫∫ r dV) / (∫∫∫ dV). Vector integration reduces the centroid computation to three scalar integrals for the x, y, and z components of the centroid position.

19. Chain Rule for Multiple Variables

The rule for differentiating composite functions of multiple variables. If f = f(x,y,z) and x, y, z are functions of t, then df/dt = (∂f/∂x)dx/dt + (∂f/∂y)dy/dt + (∂f/∂z)dz/dt = ∇f·r'(t), the dot product of the gradient and the velocity vector. This is the most important formula connecting scalar field theory to the geometry of curves.

20. Chain Rule for Vector Functions

The rule for differentiating a composition involving vector functions. If r(t) is a vector function and f is a scalar function of position, then d/dt[f(r(t))] = ∇f · r'(t). For compositions of vector functions, the chain rule applies component-wise. The chain rule for vector functions is used in computing derivatives along paths and in transforming differential operators between coordinate systems.

21. Circulation

The line integral of a vector field F around a closed curve C: Γ = ∮ F·dr. Circulation measures the tendency of the field to rotate or swirl around the loop. In fluid mechanics, circulation is related to lift by the Kutta-Joukowski theorem. By Stokes’ theorem, the circulation around a closed curve equals the flux of the curl of F through any surface bounded by the curve.

22. Circulation Theorem

The statement that the circulation of a vector field around a closed curve equals the flux of its curl through any surface bounded by that curve (Stokes’ theorem). In fluid mechanics, the Kelvin circulation theorem states that the circulation around a material loop is conserved in inviscid, barotropic flow. Circulation theorems are the basis for understanding lift generation in aerodynamics.

23. Closed Curve

A curve whose starting point and ending point coincide. Closed curves are used in the statement of the Cauchy integral theorem in complex analysis, Green’s theorem in the plane, and Stokes’ theorem in three dimensions. A simply closed curve does not cross itself. The orientation of a closed curve (counterclockwise positive by convention) determines the sign of the line integral.

24. Closed Surface

A surface that encloses a bounded region of space with no boundary of its own. A surface without a boundary (an edge). A sphere and a cube are examples. Closed surfaces are used in the statement of the Divergence (Gauss’s) theorem: the outward flux through a closed surface equals the volume integral of the divergence over the enclosed region.

25. Collinear Vectors

Vectors that lie along the same line or parallel lines, meaning they are scalar multiples of each other. Two nonzero vectors A and B are collinear if and only if A × B = 0. Collinearity is the condition for parallel forces in statics and for parallel velocity vectors in fluid flow. Collinear vectors have either zero or 180-degree angles between them.

26. Component of a Vector

The scalar projection of a vector onto a given direction, equal to the dot product of the vector with the unit vector in that direction. The component of A in the direction of unit vector û is A·û = |A|cos θ. Components are the building blocks of the Cartesian representation and allow vector equations to be converted into scalar equations along each coordinate direction.

27. Conservative Test

The procedure for determining whether a vector field F is conservative (path-independent). In a simply connected three-dimensional domain, F is conservative if and only if ∇ × F = 0. In two dimensions, F = Pi + Qj is conservative if ∂P/∂y = ∂Q/∂x. If F passes the test, the potential function f with ∇f = F is found by integrating F component by component.

28. Conservative Vector Field

A vector field F for which the line integral ∫ F·dr is path-independent — its value depends only on the endpoints of the path and not on the route taken. A vector field is conservative if and only if its curl is zero (in a simply connected domain) and equivalently if F = ∇f for some scalar potential function f. Gravity and the electrostatic force are conservative fields.

29. Continuity Equation

The vector PDE expressing conservation of mass for a fluid: ∂ρ/∂t + ∇·(ρv) = 0, where ρ is density and v is the velocity vector field. For incompressible flow (constant density), this simplifies to ∇·v = 0. The continuity equation is one of the fundamental applications of the divergence concept in engineering and is derived by applying the Divergence theorem to a fixed control volume.

30. Coordinate Surfaces

Surfaces on which one coordinate is held constant in a curvilinear coordinate system. In cylindrical coordinates, r = const gives cylinders, φ = const gives half-planes, and z = const gives horizontal planes. In spherical coordinates, r = const gives spheres, θ = const gives cones, and φ = const gives half-planes. The intersection of two coordinate surfaces gives a coordinate curve.

31. Coplanar Vectors

Three or more vectors that lie in the same plane. Three vectors A, B, C are coplanar if and only if their scalar triple product A·(B × C) = 0. Coplanarity is equivalent to linear dependence in three-dimensional space. In statics, coplanar force systems are analyzed using two-dimensional equilibrium equations rather than the full three-dimensional vector equations.

32. Covariant and Contravariant Components

In curvilinear coordinate systems, vectors have two sets of components, the covariant components that transform like coordinate differentials and contravariant components that transform like partial derivatives. In orthogonal Cartesian coordinates the two sets coincide, but in general curvilinear or non-orthogonal coordinates they differ. The distinction is essential in tensor analysis and general coordinate transformations.

33. Cross Product

The vector product A × B of two vectors, producing a vector perpendicular to both A and B with magnitude |A||B|sin θ, where θ is the angle between them. The direction is given by the right-hand rule. In Cartesian form, A × B is computed as a 3×3 determinant with i, j, k in the first row. The cross product is used to compute torques, moments, areas, and magnetic forces.

34. Cross Product Properties

Key algebraic properties of the cross product: it is anti-commutative (A × B = −B × A), distributive over addition (A × (B + C) = A × B + A × C), and not associative. The cross product of a vector with itself is zero. The cross products of the standard basis vectors satisfy i × j = k, j × k = i, k × i = j, and the reverse products give negative results.

35. Curl

The vector differential operator ∇ × F, measuring the rotational tendency or vorticity of a vector field F at a point. In Cartesian coordinates, the curl has components given by differences of partial derivatives. The curl is zero for conservative fields. By Stokes’ theorem, the surface integral of the curl over a surface equals the circulation of F around the boundary curve.

36. Curl-Free Field

See Irrotational Field. A vector field with ∇ × F = 0 everywhere. In a simply connected domain, curl-free is equivalent to conservative. The electric field in electrostatics and the gravitational field are curl-free, allowing the definition of electric potential and gravitational potential energy respectively.

37. Curl in Cylindrical Coordinates

The expression for ∇ × F in cylindrical coordinates (r, φ, z), where F = Fᵣeᵣ + Fφeφ + Fzez. Each component involves specific combinations of partial derivatives and metric scale factors. The cylindrical form is used when the problem has axial symmetry, such as in magnetic fields around current-carrying wires, flow in pipes, and heat conduction in cylindrical solids.

38. Curl in Spherical Coordinates

The expression for ∇ × F in spherical coordinates (r, θ, φ), where F = Fᵣeᵣ + Fθeθ + Fφeφ. The formula involves scale factors r and r sin θ and their combinations with partial derivatives of the field components. Spherical coordinate curl formulas are used in problems with point symmetry, such as the magnetic field of a magnetic dipole and gravitational field computations.

39. Curl of a Gradient

The identity ∇ × (∇f) = 0 for any twice continuously differentiable scalar function f. This identity states that the curl of any gradient field is identically zero. It is the vector calculus analog of the equality of mixed partial derivatives. The vanishing of the curl is a necessary condition for a vector field to be conservative (the gradient of a scalar potential).

40. Curvilinear Coordinates

Coordinate systems in which the coordinate curves are not straight lines but general curves. Cylindrical and spherical coordinates are the most important curvilinear systems in engineering. In curvilinear coordinates, the basis vectors vary in direction from point to point, and the differential operators (gradient, divergence, curl, and Laplacian) take more complex forms involving metric scale factors.

41. Curvilinear Volume Element

The volume element dV = h₁h₂h₃ dq₁ dq₂ dq₃ in a general orthogonal curvilinear coordinate system (q₁, q₂, q₃) with scale factors h₁, h₂, h₃. For cylindrical coordinates, dV = r dr dφ dz. For spherical coordinates, dV = r² sin θ dr dθ dφ. The correct volume element is essential for triple integration in non-Cartesian coordinate systems.

42. Cylindrical Coordinates

A three-dimensional coordinate system (r, φ, z) where r is the radial distance from the z-axis, φ is the azimuthal angle in the xy-plane measured from the positive x-axis, and z is the height. The conversion to Cartesian is x = r cos φ, y = r sin φ, z = z. Cylindrical coordinates are natural for problems with axial symmetry such as pipes, cylinders, and coaxial cables.

43. Del Operator

The vector differential operator ∇ = i ∂/∂x + j ∂/∂y + k ∂/∂z in Cartesian coordinates. Acting on a scalar field, ∇ gives the gradient. Acting on a vector field via the dot product, ∇· gives the divergence. Acting on a vector field via the cross product, ∇× gives the curl. Acting twice via the dot product, ∇·∇ = ∇² gives the Laplacian. The del operator is the central symbol of vector calculus.

44. Density Function

A scalar field ρ(x, y, z) assigning mass per unit volume (or mass per unit area, or mass per unit length) to each point of a body. Volume integrals of the density function over a region give the total mass, while moments of the density function give the center of mass and moments of inertia. Density functions are used in all mass-property calculations via vector integration.

45. Derivative of a Vector Function

The limit r'(t) = lim[Δt→0] [r(t+Δt) − r(t)]/Δt, computed component-wise for r(t) = x(t)i + y(t)j + z(t)k as r'(t) = x'(t)i + y'(t)j + z'(t)k. The derivative of a vector function gives the rate of change of the vector with respect to the parameter t. When r(t) is a position vector, r'(t) is the velocity vector.

46. Directed Line Segment

A line segment with a specified initial point (tail) and terminal point (head), representing a vector geometrically. The vector is determined by the displacement from tail to head. Two directed line segments with the same length and direction represent the same vector even if they have different initial points, since free vectors are characterized only by magnitude and direction.

47. Direction Angles

The angles α, β, γ that a vector A makes with the positive x-, y-, and z-axes respectively, given by cos α = Aₓ/|A|, cos β = Ay/|A|, cos γ = Az/|A|. The quantities cos α, cos β, cos γ are the direction cosines of A and satisfy cos²α + cos²β + cos²γ = 1. Direction angles provide a complete description of the orientation of a vector in three-dimensional space.

48. Direction Cosines

The cosines of the direction angles that a vector makes with the three coordinate axes, l = cos α, m = cos β, n = cos γ. They satisfy l² + m² + n² = 1 and are the components of the unit vector in the direction of A. Direction cosines are used to specify the orientation of lines and axes in three-dimensional geometry and in the transformation of vector and tensor components between coordinate systems.

49. Directional Derivative

The rate of change of a scalar function f in the direction of a unit vector û, defined as Dû f = ∇f · û = |∇f| cos θ, where θ is the angle between the gradient and û. The directional derivative is maximized when û points in the direction of ∇f and is zero when û is perpendicular to ∇f. It generalizes the partial derivative to an arbitrary direction.

50. Displacement Vector

The vector representing the change in position of a point, equal to the final position vector minus the initial position vector: Δr = r₂ − r₁. The displacement is a free vector, the initial and final positions are not important only its magnitude and direction. In mechanics, displacement is integrated over time to obtain position, and its derivative with respect to time is velocity.

51. Distance Between Two Points

The magnitude of the displacement vector between two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), given by d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]. This is the three-dimensional extension of the Pythagorean theorem. In vector notation, d = |r₂ − r₁|. Distance is a fundamental scalar quantity in geometry and is used in defining norms and metrics on vector spaces.

52. Divergence

The scalar differential operator ∇·F measuring the net outward flux of a vector field F per unit volume at a point. In Cartesian coordinates, ∇·F = ∂Fₓ/∂x + ∂Fy/∂y + ∂Fz/∂z. Positive divergence indicates a source and negative divergence indicates a sink. The Divergence theorem relates the volume integral of the divergence to the surface flux integral. Divergence is zero for incompressible flows.

53. Divergence in Cylindrical Coordinates

The expression for ∇·F in cylindrical coordinates (r, φ, z): ∇·F = (1/r)∂(rFᵣ)/∂r + (1/r)∂Fφ/∂φ + ∂Fz/∂z. The additional term involving 1/r comes from the fact that the cylindrical basis vectors change direction as φ changes. This form is used in computing the divergence of axially symmetric fields in pipe flow, rotating machinery, and electromagnetic problems.

54. Divergence in Spherical Coordinates

The expression for ∇·F in spherical coordinates (r, θ, φ): ∇·F = (1/r²)∂(r²Fᵣ)/∂r + (1/r sin θ)∂(sin θ Fθ)/∂θ + (1/r sin θ)∂Fφ/∂φ. The scale factors reflect the geometry of the spherical coordinate system. This form is used in gravitational field analysis, electrostatics with spherical symmetry, and acoustic radiation problems.

55. Divergence of a Curl

The identity ∇·(∇ × F) = 0 for any twice continuously differentiable vector field F. This identity states that the divergence of any curl field is identically zero. It is the vector calculus analog of the second mixed partial equality and is used to check consistency of equations in electromagnetic theory, where it implies that magnetic monopoles do not exist.

56. Divergence Theorem

The theorem, also known as Gauss’s theorem, stating that the volume integral of the divergence of a vector field F over a region V equals the outward flux through the closed bounding surface S: ∫∫∫_V ∇·F dV = ∬_S F·n dS. It converts between volume and surface integrals and is the mathematical expression of conservation laws in field theory, fluid mechanics, and electrostatics.

57. Dot Product

The scalar product A·B = |A||B|cos θ = AₓBₓ + AyBy + AzBz, where θ is the angle between the two vectors. The dot product is commutative (A·B = B·A), distributive over addition, and produces a scalar. It is used to compute work done by a force, projections, angles between vectors, and to test perpendicularity. Two vectors are perpendicular if and only if their dot product is zero.

58. Dot Product Properties

Key algebraic properties: the dot product is commutative, bilinear, and positive definite (A·A = |A|² ≥ 0, with equality only when A = 0). For orthonormal basis vectors, i·i = j·j = k·k = 1 and i·j = j·k = k·i = 0. The dot product satisfies the Cauchy-Schwarz inequality |A·B| ≤ |A||B|, which is the vector form of the classical inequality.

59. Double Integral

The integral of a scalar function f(x,y) over a two-dimensional region R in the plane, written ∬_R f(x,y) dA. Double integrals compute areas (when f = 1), volumes under surfaces, masses of planar laminae, and other accumulated quantities over plane regions. They are evaluated by iterated integration, and the order of integration can be switched using Fubini’s theorem when f is continuous.

60. Electric Field Vector

A vector field E(r) representing the force per unit charge that a positive test charge would experience at each point in space. In electrostatics, E = −∇V where V is the electric potential. The electric field is related to charge distributions by Gauss’s law: ∇·E = ρ/ε₀. Field lines run from positive to negative charges and are everywhere tangent to E.

61. Equipotential Surface

A surface on which a scalar potential function f has a constant value. Equipotential surfaces are always perpendicular to the gradient vector ∇f. In electrostatics, they are surfaces of constant electric potential and are perpendicular to the electric field lines. In heat conduction, they are isothermal surfaces perpendicular to the heat flux vector. No work is done moving along an equipotential surface in a conservative field.

62. Euler’s Equations of Motion

The vector equations governing the rotational motion of a rigid body about a fixed point, derived from the angular momentum theorem. In component form, they involve the moments of inertia and the angular velocity components. Euler’s equations are the fundamental vector equations in rotational dynamics and require the use of body-fixed reference frames where the inertia tensor is constant.

63. Field Line

A curve in a vector field whose tangent at every point is parallel to the field vector at that point. Field lines visualize the direction and (by their spacing) the magnitude of a vector field. Electric field lines originate on positive charges and terminate on negative charges. Streamlines in fluid mechanics are the field lines of the velocity field. Field lines never cross in a region where the field is nonzero.

64. Flux

The surface integral of the normal component of a vector field F through a surface S: Φ = ∬_S F·n dS, where n is the outward unit normal. Flux measures the total amount of the field passing through the surface per unit time. In fluid mechanics, it is the volume flow rate. In electrostatics, the total electric flux through a closed surface equals the enclosed charge divided by ε₀ (Gauss’s law).

65. Flux Integral

Another term for the surface integral of the normal component of a vector field, ∬_S F·dS = ∬_S F·n dS. The flux integral is the primary quantity in the Divergence theorem and in Faraday’s law and Gauss’s law in electromagnetic theory. Its evaluation requires choosing a consistent normal direction (inward or outward) and parametrizing the surface.

66. Free Vector

A vector that is not associated with any particular point of application and can be freely translated in space without changing its meaning. Displacement vectors, velocities, and forces (in the context of finding their net resultant) are treated as free vectors. In contrast, bound vectors or sliding vectors have fixed or restricted lines of action.

67. Frenet-Serret Formulas

The differential equations dT/ds = κN, dN/ds = −κT + τB, dB/ds = −τN, relating the rates of change of the unit tangent T, principal normal N, and binormal B with respect to arc length s. Here κ is the curvature and τ is the torsion of the curve. The Frenet-Serret formulas completely describe the local geometry of a smooth space curve at each point.

68. Fundamental Theorem for Line Integrals

The theorem stating that the line integral of the gradient of a scalar function f along a curve C from point A to point B equals f(B) − f(A), regardless of the path taken. It is the vector calculus generalization of the fundamental theorem of calculus and shows that line integrals of conservative fields are path-independent and determined entirely by the potential function values at the endpoints.

69. Gauss’s Divergence Theorem

See Divergence Theorem. Named for Carl Friedrich Gauss, this theorem is the three-dimensional analog of Green’s theorem and one of the fundamental results of vector integral calculus. It converts volume integrals of the divergence into surface integrals of the flux and is applied in deriving conservation laws, computing electric flux, and simplifying volume integrals in engineering problems.

70. Gauss’s Law

In electrostatics, the statement that the total outward electric flux through any closed surface equals the total enclosed charge divided by the permittivity of free space: ∬_S E·n dS = Q_enc/ε₀. Gauss’s law follows from Coulomb’s law and the Divergence theorem. It is most useful for computing electric fields of symmetric charge distributions — spheres, cylinders, and infinite planes.

71. Geometric Vector

A vector defined geometrically as a directed line segment, characterized by magnitude (length) and direction (orientation in space) but not by position. Two directed line segments are considered the same geometric vector if they have equal length and parallel direction. The geometric view of vectors is the foundation for the parallelogram and triangle laws of vector addition.

72. Gradient

The vector differential operator ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k applied to a scalar function f. The gradient points in the direction of the steepest increase of f, and its magnitude equals the maximum rate of change of f at that point. The gradient is perpendicular to the level surfaces of f. It appears in Fourier’s law of heat conduction, Darcy’s law, and diffusion equations.

73. Gradient in Cylindrical Coordinates

The expression for ∇f in cylindrical coordinates (r, φ, z): ∇f = (∂f/∂r)eᵣ + (1/r)(∂f/∂φ)eφ + (∂f/∂z)ez. The factor 1/r in the φ-component accounts for the fact that a unit change in φ corresponds to an arc length of r, not one unit. This form is used when scalar fields have cylindrical symmetry, such as the temperature distribution in a cylindrical rod.

74. Gradient in Spherical Coordinates

The expression for ∇f in spherical coordinates (r, θ, φ): ∇f = (∂f/∂r)eᵣ + (1/r)(∂f/∂θ)eθ + (1/r sin θ)(∂f/∂φ)eφ. The scale factors 1/r and 1/(r sin θ) reflect the geometry of spherical coordinates. This form is used for fields with spherical symmetry, such as the gravitational potential of a point mass or the electric potential of a spherically symmetric charge distribution.

75. Gradient of a Vector Field

The tensor (dyadic) generalization of the scalar gradient, defined as ∇F with components ∂Fᵢ/∂xⱼ forming a 3×3 matrix. The gradient of a vector field is a second-order tensor (the Jacobian matrix) and appears in the constitutive relations of fluid mechanics (the viscous stress tensor involves the gradient of the velocity field). It is not the same as the vector del operator applied to a scalar.

76. Gradient Operator

The vector differential operator ∇ = i ∂/∂x + j ∂/∂y + k ∂/∂z that, when applied to a scalar function f, produces the gradient vector ∇f. It is the key operator that converts potential energy functions to force fields, temperature fields to heat flux vectors, and electric potential to electric field. The gradient operator points in the direction of greatest increase of the scalar field.

77. Gradient Theorem

See Fundamental Theorem for Line Integrals. The gradient theorem is the precise statement that conservative vector fields (gradients of scalar functions) have path-independent line integrals. It connects the local property of a field being a gradient to the global property of path independence. The theorem is the starting point for potential theory in both fluids and electrostatics.

78. Green’s First Identity

The integral identity ∫∫∫_V u∇²v dV = ∬_S u(∇v·n) dS − ∫∫∫_V ∇u·∇v dV, derived by applying the Divergence theorem to the product u∇v. It is used in proving uniqueness theorems for Laplace’s and Poisson’s equations, in the derivation of the boundary element method, and in establishing reciprocity relations between pairs of solutions to elliptic PDEs.

79. Green’s Second Identity

The symmetric identity ∫∫∫_V (u∇²v − v∇²u) dV = ∬_S (u∇v − v∇u)·n dS, obtained from Green’s first identity by interchanging u and v and subtracting. It is used to prove uniqueness for the Dirichlet and Neumann problems, to construct Green’s functions for the Laplacian, and to derive the representation formula for harmonic functions in terms of their boundary values.

80. Green’s Theorem

The theorem relating the line integral around a closed curve C in the plane to the double integral over the enclosed region D: ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA. It is the two-dimensional special case of Stokes’ theorem. Green’s theorem is used to compute areas, evaluate line integrals by converting to double integrals, and prove results in planar potential theory.

81. Green’s Theorem — Area Formula

The application of Green’s theorem to compute the area of a region D enclosed by a curve C. Using P = 0, Q = x or P = −y, Q = 0 or P = −y/2, Q = x/2 gives Area = ∮_C x dy = −∮_C y dx = (1/2) ∮_C (x dy − y dx). This formula is particularly useful for computing areas bounded by parametric curves.

82. Hamilton’s Del

Another name for the del operator ∇, named after the Irish mathematician William Rowan Hamilton who introduced it as part of his quaternion calculus. Hamilton’s del is the symbolic vector ∇ = i ∂/∂x + j ∂/∂y + k ∂/∂z that generates the gradient, divergence, curl, and Laplacian through its different types of application to scalar and vector fields.

83. Harmonic Function

A scalar function f satisfying Laplace’s equation ∇²f = 0 throughout a domain. The real and imaginary parts of any analytic complex function are harmonic. Harmonic functions have the mean value property, satisfy the maximum principle, and are infinitely differentiable within their domain. They model steady-state temperature, electrostatic potential, gravitational potential, and ideal fluid flow.

84. Helix

A space curve traced on the surface of a cylinder, defined parametrically by r(t) = a cos(t)i + a sin(t)j + btk, where a is the radius and b controls the pitch. The helix has constant curvature κ = a/(a² + b²) and constant torsion τ = b/(a² + b²). It is one of the simplest non-planar curves and serves as a standard example in Frenet-Serret frame computations.

85. Helmholtz Decomposition Theorem

The theorem stating that any sufficiently smooth vector field F can be uniquely decomposed into the sum of an irrotational part (gradient of a scalar potential) and a solenoidal part (curl of a vector potential): F = −∇φ + ∇ × A. The Helmholtz decomposition is fundamental in electromagnetic theory, where it separates the electric field into its conservative and induced components.

86. Higher-Order Partial Derivatives

Partial derivatives taken more than once with respect to one or more variables. The mixed partial derivatives ∂²f/∂x∂y and ∂²f/∂y∂x are equal when f has continuous second partial derivatives (Clairaut’s theorem). Higher-order partial derivatives appear in the Taylor expansion of scalar fields, in the classification of PDEs, and in the study of the Laplacian and biharmonic operators.

87. Irrotational Field

A vector field F with zero curl everywhere: ∇ × F = 0. In a simply connected domain, an irrotational field is conservative and can be written as F = ∇f for some scalar potential f. Irrotational flow is called potential flow in fluid mechanics. The electric field in electrostatics is irrotational (in the absence of time-varying magnetic fields), confirming the existence of the electric potential V.

88. Irrotational Flow

Fluid flow in which the velocity field v satisfies ∇ × v = 0, meaning there is no local rotation of fluid elements. Irrotational flow is also called potential flow because the velocity can be derived from a velocity potential φ: v = ∇φ. Laplace’s equation ∇²φ = 0 governs the velocity potential for incompressible irrotational flow. This model is used in aerodynamics and hydrodynamics for ideal (inviscid) flows.

89. Jacobian Determinant

The determinant of the Jacobian matrix of a coordinate transformation, used in changing variables in multiple integrals. For the transformation from (u, v) to (x, y), the Jacobian J = ∂(x, y)/∂(u, v) is the determinant of the 2×2 matrix of partial derivatives. In cylindrical coordinates, J = r; in spherical coordinates, J = r² sin θ. The volume element transforms as dV = |J| du dv dw.

90. Jacobian Matrix

The matrix of all first-order partial derivatives of a vector-valued function F: Rⁿ → Rᵐ, with entry (i,j) equal to ∂Fᵢ/∂xⱼ. The Jacobian matrix is the linear approximation to F at a point and generalizes the derivative to vector functions. Its determinant (the Jacobian determinant) appears in change-of-variable formulas for multiple integrals and in the linearization of nonlinear systems.

91. Kinetic Energy via Vectors

The scalar quantity T = (1/2)m|v|² = (1/2)m(v·v), where v is the velocity vector and m is the mass. In rotational dynamics, the kinetic energy is T = (1/2)ω·Iω, where ω is the angular velocity vector and I is the inertia tensor. Vector forms of kinetic energy are used in Lagrangian mechanics and in energy methods for analyzing the motion of mechanical systems.

92. Laplacian

The scalar differential operator ∇²f = ∇·(∇f) = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z² applied to a scalar function f. It measures the difference between the value of f at a point and the average of f over a small surrounding sphere. The Laplacian appears in the heat equation, wave equation, Laplace’s equation, and Poisson’s equation. For a vector field F, the vector Laplacian is ∇²F = ∇(∇·F) − ∇×(∇×F).

93. Laplacian in Cylindrical Coordinates

The expression ∇²f = (1/r)∂/∂r(r ∂f/∂r) + (1/r²)∂²f/∂φ² + ∂²f/∂z² in cylindrical coordinates. This form of the Laplacian appears in the heat equation, diffusion equation, and Laplace’s equation when the geometry has axial symmetry. The separation of variables in cylindrical coordinates leads to Bessel’s equation for the radial part of the solution.

94. Laplacian in Spherical Coordinates

The expression ∇²f = (1/r²)∂/∂r(r²∂f/∂r) + (1/r²sin θ)∂/∂θ(sin θ ∂f/∂θ) + (1/r²sin²θ)∂²f/∂φ² in spherical coordinates. This form appears in problems with spherical symmetry and leads to Legendre’s equation for the angular part when variables are separated. It is used in gravitational potential theory, electrostatics, and quantum mechanics.

95. Level Curve

A curve in the xy-plane along which a scalar function f(x,y) has a constant value: f(x,y) = c. Level curves are two-dimensional analogs of level surfaces. The gradient ∇f is always perpendicular to the level curves of f. In topography, level curves are contour lines. In electrostatics, they are equipotential lines. In fluid mechanics, they are streamlines for two-dimensional flows with a stream function.

96. Level Surface

A surface in three-dimensional space along which a scalar function f(x, y, z) has a constant value: f(x, y, z) = c. The gradient ∇f is everywhere perpendicular to the level surfaces of f. Level surfaces are used to visualize scalar fields and to define the normal vector to a surface for use in surface integration. Isothermal surfaces, equipotential surfaces, and isobaric surfaces are all level surfaces.

97. Levi-Civita Symbol

The completely antisymmetric symbol εᵢⱼₖ equal to +1 for even permutations of (1,2,3), −1 for odd permutations, and 0 if any two indices repeat. Used in index notation to write the cross product as (A × B)ᵢ = εᵢⱼₖ AⱼBₖ, the curl as (∇ × F)ᵢ = εᵢⱼₖ ∂Fₖ/∂xⱼ, and the determinant in compact form. Together with the Kronecker delta, it satisfies the key identity εᵢⱼₖεᵢₘₙ = δⱼₘδₖₙ − δⱼₙδₖₘ.

98. Line Integral of a Scalar Field

The integral of a scalar function f along a curve C, defined as ∫_C f ds, where ds is the arc length element. It generalizes the ordinary integral to curved paths and is used to compute the mass of a wire with variable density, the work done against a scalar resistance, and path-dependent physical quantities. The value depends on the curve C but not on its orientation.

99. Line Integral of a Vector Field

The integral ∫_C F·dr = ∫_C (Fₓdx + Fydy + Fzdz), measuring the work done by the force field F along the curve C. It is evaluated by parametrizing the curve and substituting. For conservative fields, it depends only on the endpoints. The line integral of a vector field is the central quantity in the work-energy theorem and in the statement of Stokes’ theorem.

100. Line of Action

The infinite straight line along which a force vector lies. The line of action determines the moment (torque) of the force about any reference point and is critical in determining whether the force will cause rotation. Two forces with the same line of action produce the same moment regardless of where along the line they are applied, this is the principle of transmissibility in statics.

[ihc-hide-content ihc_mb_type=”show” ihc_mb_who=”2,3,4,5″ ihc_mb_template=”3″ ]

101. Linear Combination of Vectors

A vector of the form c₁A + c₂B + … + cₙAₙ, where c₁, c₂, …, cₙ are scalars. Any vector in the plane spanned by a set of vectors can be expressed as a linear combination of those vectors. The coefficients are found by solving a system of linear equations. Linear combinations are the building blocks of vector spaces and the mechanism behind component decomposition.

102. Linear Dependence

A set of vectors {A₁, A₂, …, Aₙ} is linearly dependent if there exist scalars c₁, c₂, …, cₙ, not all zero, such that c₁A₁ + c₂A₂ + … + cₙAₙ = 0. Linear dependence means at least one vector in the set can be written as a linear combination of the others. In three dimensions, more than three vectors are always linearly dependent, and three coplanar vectors are linearly dependent.

103. Linear Independence

A set of vectors {A₁, A₂, …, Aₙ} is linearly independent if the only solution to c₁A₁ + c₂A₂ + … + cₙAₙ = 0 is c₁ = c₂ = … = cₙ = 0. Linearly independent vectors do not lie in a lower-dimensional subspace. The maximum number of linearly independent vectors in three-dimensional space is three, and any such maximal set is a basis for the space.

104. Magnitude of a Vector

The length or norm of a vector A, denoted |A| or ‖A‖, computed as |A| = √(Aₓ² + Ay² + Az²). The magnitude is always a non-negative scalar and equals zero only for the zero vector. The magnitude satisfies |cA| = |c||A| for any scalar c and the triangle inequality |A + B| ≤ |A| + |B|. It is used to normalize vectors and to compute distances and speeds.

105. Mass via Vector Integration

The total mass of a body computed by integrating the density function over the region: m = ∫∫∫_V ρ(x,y,z) dV for a solid, or m = ∬_S ρ(x,y,z) dS for a surface, or m = ∫_C ρ ds for a wire. Vector integration allows mass computation for bodies with continuously varying density by accumulating infinitesimal mass contributions over the entire body.

106. Maxwell’s Equations

The four fundamental vector PDEs governing all classical electromagnetic phenomena, written in differential form as: ∇·E = ρ/ε₀ (Gauss’s law), ∇·B = 0 (no magnetic monopoles), ∇×E = −∂B/∂t (Faraday’s law), and ∇×B = μ₀J + μ₀ε₀∂E/∂t (Ampère-Maxwell law). Maxwell’s equations are the supreme engineering application of vector differential calculus and the foundation of electromagnetic field theory.

107. Metric Coefficients

The scaling factors hᵢ that relate coordinate differentials to physical arc length elements in a curvilinear coordinate system. In cylindrical coordinates, h₁ = 1 (r-direction), h₂ = r (φ-direction), h₃ = 1 (z-direction). In spherical coordinates, h₁ = 1, h₂ = r, h₃ = r sin θ. Metric coefficients appear in all expressions for gradient, divergence, curl, and Laplacian in non-Cartesian coordinates.

108. Mixed Triple Product

See Scalar Triple Product. The mixed triple product A·(B × C) is a scalar equal to the volume of the parallelepiped defined by the three vectors A, B, C. It is also computed as the 3×3 determinant with A, B, C as rows. The mixed triple product is zero if and only if the three vectors are coplanar.

109. Moment of a Force

The torque or moment M = r × F produced by a force F applied at position r relative to a reference point. The moment vector is perpendicular to the plane containing r and F, with magnitude |r||F|sin θ. Moments represent the rotational effect of forces and are central to statics, structural analysis, and rigid body dynamics. The vector form automatically gives the axis and sense of rotation.

110. Moment of Inertia Tensor

A second-order tensor Iᵢⱼ representing the resistance of a rigid body to angular acceleration about any axis. Its diagonal components are the principal moments of inertia and its off-diagonal components are the products of inertia. The angular momentum L and angular velocity ω are related by L = Iω. Diagonalizing the inertia tensor gives the principal axes and principal moments of inertia.

111. Moving Trihedral

The moving frame consisting of the unit tangent T, principal normal N, and binormal B vectors attached to a point on a space curve and moving along with it as the point traverses the curve. Also called the Frenet-Serret frame. The moving trihedral provides a local coordinate system aligned with the geometry of the curve and is used in the study of curves in space and in dynamics on curved paths.

112. Nabla

Another name for the del operator ∇. The symbol ∇ is read as “nabla” or “del.” As a gradient operator it acts on scalars, as a divergence operator it acts on vectors via dot product, and as a curl operator it acts on vectors via cross product. The name “nabla” comes from a Hebrew or Greek word for a type of harp, suggested by the shape of the symbol.

113. Net Flux

The total outward flux of a vector field through a closed surface, equal to the sum of all outward contributions minus all inward contributions. By the Divergence theorem, the net flux through a closed surface equals the volume integral of the divergence over the enclosed region. Net flux is positive if there is a net source inside the surface and negative if there is a net sink.

114. Newton’s Law in Vector Form

The second law of motion written as F = ma, where F is the net force vector, m is the scalar mass, and a is the acceleration vector. The vector form automatically accounts for direction. The component form gives three scalar equations, one along each coordinate axis. Newton’s law in vector form is the foundation of classical mechanics and the starting point for all equations of motion.

115. Normal Plane

The plane through a point on a curve that is perpendicular to the tangent vector at that point. The normal plane contains both the principal normal N and the binormal B and represents all directions perpendicular to the direction of travel along the curve. The normal plane is used in the geometric description of curves and in the study of intersections of curves with surfaces.

116. Normal to a Surface

A vector perpendicular to the tangent plane of a surface at a given point. For a surface defined by F(x,y,z) = 0, the normal direction is ∇F. For a parametric surface r(u,v), the normal is ∂r/∂u × ∂r/∂v. The unit normal n = N/|N| is used in surface integrals, flux calculations, and in defining the orientation of a surface for the application of Stokes’ and Gauss’s theorems.

117. Normal Vector

A vector perpendicular to a given surface, curve (in 2D), or plane at a specified point. For the plane ax + by + cz = d, the normal vector is (a, b, c). For a curve r(t), the principal normal N = T’/|T’| points toward the center of curvature. Unit normal vectors are required for surface integrals and for applying the Divergence and Stokes’ theorems.

118. Null Vector

The zero vector 0, which has zero magnitude and no defined direction. The zero vector is the additive identity for vector addition: A + 0 = A for any vector A. It is the only vector for which the angle with another vector is undefined. The cross product of any vector with itself equals the zero vector: A × A = 0.

119. Oriented Surface

A surface S for which a consistent choice of normal direction has been made at every point. For a closed surface, the outward normal is the standard positive orientation. For an open surface, the orientation is tied to the boundary curve’s orientation by the right-hand rule in Stokes’ theorem. Orientation is essential for the sign of surface integrals and for the correct application of the integral theorems.

120. Orthogonal Coordinates

A curvilinear coordinate system in which the three families of coordinate surfaces are mutually perpendicular at every point. Cylindrical and spherical coordinates are orthogonal. In orthogonal coordinates, the metric tensor is diagonal and the differential operators simplify compared to non-orthogonal systems. The basis vectors eᵢ are mutually perpendicular unit vectors that vary in direction from point to point.

121. Orthogonal Projection

The projection of a vector A onto a direction û given by (A·û)û. The orthogonal projection removes the component perpendicular to û and retains only the component parallel to û. Projections are used in decomposing forces, velocities, and fields into components along specified directions. The vector A, (A·û)û is the component of A perpendicular to û.

122. Osculating Circle

The circle of best fit to a curve at a given point. The circle that has the same tangent and the same curvature as the curve at that point. Its radius is the radius of curvature R = 1/κ and its center is the center of curvature. The osculating circle lies in the osculating plane (spanned by T and N) and characterizes the second-order geometry of the curve at the point.

123. Osculating Plane

The plane containing the unit tangent T and the principal normal N at a point on a space curve. The osculating plane is the plane of best fit to the curve at that point and contains the osculating circle. For a planar curve, the osculating plane coincides with the plane of the curve everywhere. The binormal B is perpendicular to the osculating plane.

124. Outward Normal

The unit normal vector pointing away from the interior of a closed surface or region, used as the positive orientation for applying the Divergence theorem. The choice of outward (rather than inward) normal is the conventional positive orientation for closed surfaces. Reversing the normal direction reverses the sign of the surface integral.

125. Parallel Vectors

Two nonzero vectors A and B are parallel if A = cB for some nonzero scalar c. Parallel vectors have zero cross product: A × B = 0. They are either in the same direction (c > 0) or opposite directions (c < 0). In engineering, parallel forces can be added algebraically if they act along the same line, and their resultant acts along the same line.

126. Parametric Curve

A curve defined by specifying the position vector as a function of a parameter t: r(t) = x(t)i + y(t)j + z(t)k. The parameter t typically represents time or arc length. Parametric curves are the standard representation for computing line integrals, tangent vectors, curvature, and torsion. They can represent curves that cannot be expressed as explicit functions y = f(x).

127. Parametric Surface

A surface defined by a position vector function of two parameters: r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k. The partial derivatives ∂r/∂u and ∂r/∂v are tangent vectors to the surface, and their cross product gives the surface normal. Parametric surfaces are the standard form for computing surface integrals and for evaluating flux through curved surfaces.

128. Path Independence

The property of a line integral ∫_C F·dr such that the value depends only on the starting and ending points of the path C and not on the specific route taken. Path independence is equivalent to the vector field F being conservative (having zero curl in a simply connected domain). It is equivalent to the existence of a scalar potential function f such that F = ∇f.

129. Planar Curve

A curve that lies entirely in a single plane. For a planar curve, the torsion is identically zero everywhere. Circles, ellipses, parabolas, and all conic sections are planar curves. The Frenet-Serret formulas for planar curves reduce to just the tangent and normal vectors, with the binormal being constant and perpendicular to the plane of the curve.

130. Polar Coordinates

A two-dimensional coordinate system (r, θ) where r is the radial distance from the origin and θ is the angle from the positive x-axis. Polar coordinates are useful for regions with circular symmetry. The area element in polar coordinates is r dr dθ. The gradient and Laplacian in polar coordinates are special cases of the cylindrical formulas with z set to a constant.

131. Position Vector

The vector r = xi + yj + zk drawn from the origin to the point (x, y, z) in three-dimensional space. The position vector completely specifies the location of a point. Derivatives of the position vector with respect to time give the velocity and acceleration vectors. In line integrals, the differential dr = dx i + dy j + dz k of the position vector is the path element.

132. Potential Energy via Vector Fields

The scalar function V(r) associated with a conservative force field F through the relation F = −∇V. The potential energy decreases in the direction of the force. The work done by the force in moving from point A to B equals V(A) − V(B), confirming energy conservation. In electrostatics, V is the electric potential and F = qE = −q∇V is the force on a charge q.

133. Principal Normal Vector

The unit vector N = (dT/ds)/|dT/ds| = T’/|T’|, pointing from the curve toward the center of curvature. The principal normal is always perpendicular to the tangent T and lies in the osculating plane. Together with T and the binormal B, it forms the Frenet-Serret moving trihedral. The principal normal indicates the direction in which the curve is bending at each point.

134. Product Rules for Vector Derivatives

The set of differentiation rules for products involving scalar and vector functions. Key rules include: d/dt(f A) = f’A + fA’, d/dt(A·B) = A’·B + A·B’, and d/dt(A × B) = A’ × B + A × B’. These rules have the same form as the scalar product rule but order must be preserved in the cross product since it is not commutative.

135. Projection of a Vector

The component of a vector A in a given direction, computed as (A·û)û for the vector projection or A·û for the scalar projection, where û is the unit vector in the desired direction. The scalar projection gives the signed length of the projection. Projections are used to decompose forces and velocities, to compute work, and to find the shortest distance from a point to a plane.

136. Radius of Curvature

The reciprocal of the curvature, R = 1/κ, representing the radius of the osculating circle at a point on a curve. A large radius of curvature means the curve is nearly straight, while a small radius means it is sharply curved. The radius of curvature appears in the centripetal acceleration formula a_N = v²/R and is used in road design, beam bending theory, and optical system analysis.

137. Rectifying Plane

The plane through a point on a space curve containing the unit tangent T and the binormal B. The rectifying plane is perpendicular to the principal normal N and is the plane in which the curve has zero curvature (it appears straightest). Together with the osculating plane and the normal plane, the rectifying plane completes the set of three fundamental planes of the Frenet-Serret frame.

138. Resultant Vector

The single vector equivalent to the combined effect of a system of vectors, obtained by vector addition. The resultant of forces F₁, F₂, …, Fₙ is R = F₁ + F₂ + … + Fₙ. In equilibrium, the resultant equals the zero vector. The resultant is found graphically by the polygon rule or analytically by summing components along each coordinate axis.

139. Right-Hand Rule

The convention for determining the direction of the cross product A × B or the curl of a field: curl the fingers of the right hand from A toward B (through the smaller angle), and the extended thumb points in the direction of A × B. The right-hand rule also determines the positive normal to an oriented surface. Curl the fingers of the right hand around the surface in the direction of the boundary curve and the thumb indicates the positive normal.

140. Rotation

In vector analysis, a rotation is represented by the angular velocity vector ω, where its direction is the axis of rotation (by the right-hand rule) and its magnitude is the angular speed in radians per second. The velocity of a point at position r relative to the axis in a rotating rigid body is v = ω × r. Rotations are non-commutative in general but infinitesimal rotations commute and behave as vectors.

141. Scalar

A quantity completely described by a single real number with appropriate units, having magnitude but no direction. Temperature, density, pressure, mass, and electric potential are scalars. Scalars are invariant under rotation of the coordinate system. They will have the same value regardless of how the axes are oriented. Scalars are rank-0 tensors.

142. Scalar Field

A function that assigns a scalar value to every point in a region of space, f: R³ → R. Temperature distribution T(x,y,z), pressure P(x,y,z), and electric potential V(x,y,z) are scalar fields. The gradient of a scalar field is a vector field. Level surfaces of a scalar field are the surfaces on which the field is constant and are everywhere perpendicular to the gradient.

143. Scalar Triple Product

The scalar A·(B × C), equal to the determinant of the 3×3 matrix with rows A, B, and C. Its absolute value equals the volume of the parallelepiped with edges along A, B, and C. The scalar triple product is zero if and only if the three vectors are coplanar. It satisfies the cyclic property A·(B × C) = B·(C × A) = C·(A × B).

144. Simply Connected Domain

A region in which every closed curve can be continuously shrunk to a point without leaving the region. A simply connected domain has no holes. In a simply connected domain, an irrotational vector field (curl = 0) is guaranteed to be conservative (gradient of a scalar). The condition of simple connectivity is essential for the equivalence between zero curl and path independence.

145. Sink

A point or region in a vector field where the divergence is negative, meaning field lines converge into the region. In fluid mechanics, a sink absorbs fluid at a given rate. In the two-dimensional complex potential framework, a point sink has logarithmic potential with negative strength. The Divergence theorem relates the volume integral of divergence (summing sources and sinks) to the net outward flux.

146. Smooth Curve

A curve parametrized by r(t) where r'(t) is continuous and nonzero on the parameter interval. Smoothness ensures the curve has a well-defined tangent direction at every point. A piecewise smooth curve consists of a finite number of smooth arcs joined end to end. Line integrals are defined along piecewise smooth curves, which is sufficient for almost all engineering applications.

147. Smooth Surface

A surface parametrized by r(u,v) where the partial derivatives ∂r/∂u and ∂r/∂v are continuous and their cross product is nonzero. Smoothness ensures the surface has a well-defined tangent plane and normal vector at every point. A piecewise smooth surface consists of finitely many smooth pieces. Surface integrals are defined on piecewise smooth surfaces, which covers all standard engineering geometries.

148. Solenoidal Field

A vector field F with zero divergence everywhere: ∇·F = 0. Solenoidal fields have no sources or sinks. Thus, all field lines are closed loops or extend to infinity without terminating. The magnetic field B is always solenoidal (∇·B = 0). By the Helmholtz decomposition theorem, every solenoidal field can be written as the curl of a vector potential: F = ∇ × A.

149. Source

A point or region in a vector field where the divergence is positive, meaning field lines originate and flow outward from that region. A positive electric charge is a source of the electric field. In fluid mechanics, a source injects fluid at a given rate. Sources and sinks can be superposed in potential flow theory to model complex flows around objects.

150. Space Curve

A curve that does not lie in any single plane, traced out by the tip of a position vector r(t) as t varies. Space curves have both curvature and torsion; the curvature measures bending and torsion measures twisting. The helix is the simplest space curve with constant curvature and torsion. Space curves are analyzed using the Frenet-Serret frame and the moving trihedral.

151. Span of a Set of Vectors

The set of all linear combinations of the vectors in the set. The span of {A, B} is the plane (or line, if parallel) containing all vectors of the form αA + βB. If A, B, C are linearly independent vectors in R³, their span is all of R³. The concept of span is fundamental to understanding vector spaces, bases, and the range of linear transformations.

152. Speed

The magnitude of the velocity vector: v = |v| = |dr/dt|. Speed is a non-negative scalar, while velocity is a vector. On a parametric curve, the speed at parameter t is |r'(t)|, which is the integrand in the arc length formula. Distinguishing speed (scalar) from velocity (vector) is essential in mechanics and is one of the first conceptual distinctions made in vector analysis courses.

153. Spherical Coordinates

A three-dimensional coordinate system (r, θ, φ) where r is the distance from the origin, θ is the polar angle from the positive z-axis (colatitude), and φ is the azimuthal angle from the positive x-axis. Conversion to Cartesian: x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. Spherical coordinates are natural for problems with point symmetry such as gravitational fields and antenna radiation patterns.

154. Stokes’ Theorem

The theorem relating the surface integral of the curl of a vector field F over a surface S to the line integral of F around the boundary curve C: ∬_S (∇ × F)·n dS = ∮_C F·dr. The boundary curve C must be oriented consistently with the surface normal by the right-hand rule. Stokes’ theorem reduces surface integrals to line integrals and is used in electromagnetic theory and fluid mechanics.

155. Streamline

A curve in a fluid flow field that is everywhere tangent to the velocity vector at a given instant in time. For steady flow, streamlines, pathlines (trajectories of particles), and streaklines coincide. The stream function ψ is constant along streamlines in two-dimensional incompressible flow. Streamlines never cross (except at stagnation points) and provide a visual map of the instantaneous flow pattern.

156. Stress Tensor

A second-order tensor σᵢⱼ representing the internal forces per unit area (stresses) within a deformable body. The stress vector (traction) on a surface with unit normal n is T = σn. The diagonal components are normal stresses and the off-diagonal components are shear stresses. The stress tensor is symmetric for a body in equilibrium and is the key quantity in continuum mechanics and structural analysis.

157. Subspace

A subset of a vector space that is itself a vector space under the same operations. A subspace must contain the zero vector, be closed under addition, and be closed under scalar multiplication. The span of any set of vectors is a subspace. In R³, the subspaces are the zero vector, all lines through the origin, all planes through the origin, and R³ itself.

158. Surface

A two-dimensional manifold in three-dimensional space, described by a parametric representation r(u,v), an implicit equation F(x,y,z) = 0, or an explicit equation z = f(x,y). Surfaces are the domains for surface integrals and are the boundaries of three-dimensional regions in the Divergence theorem. Standard surfaces in engineering include spheres, cylinders, cones, paraboloids, and flat planes.

159. Surface Area

The total area of a surface S, computed as ∬_S dS = ∬_D |∂r/∂u × ∂r/∂v| du dv for a parametric surface r(u,v) over a parameter domain D. For an explicit surface z = f(x,y), the area element is dS = √(1 + (∂f/∂x)² + (∂f/∂y)²) dA. Surface area calculations arise in heat transfer, fluid pressure on curved surfaces, and structural coating and loading problems.

160. Surface Integral of a Scalar Field

The integral ∬_S f(x,y,z) dS of a scalar function over a surface S, where dS is the scalar area element. It computes the total mass of a thin shell with surface density f, the center of mass of a shell, and other surface-distributed quantities. The surface integral of a scalar field does not depend on the orientation of the surface.

161. Surface Integral of a Vector Field

The flux integral ∬_S F·n dS = ∬_S F·dS, measuring the net flow of the vector field F through the oriented surface S. It depends on the choice of normal orientation. Surface integrals of vector fields are the key quantities in the Divergence theorem, Stokes’ theorem, Faraday’s law, and Gauss’s law. They require both a parametrization of the surface and a consistent normal direction.

162. Tangent Plane

The plane that best approximates a smooth surface at a given point, spanned by the tangent vectors ∂r/∂u and ∂r/∂v at that point. For the surface F(x,y,z) = 0, the equation of the tangent plane at (x₀,y₀,z₀) is ∇F·(r − r₀) = 0, confirming that the gradient is normal to the surface. The tangent plane is used in linearization, surface integrals, and the definition of smooth surfaces.

163. Tangent Vector

The derivative of the position vector with respect to the parameter: r'(t) = dr/dt. The tangent vector points in the direction of travel along the curve and has magnitude equal to the speed. The unit tangent vector T = r'(t)/|r'(t)| provides the direction of the curve at each point. Tangent vectors are used to compute line integrals, define the Frenet-Serret frame, and determine the direction of the curve at any point.

164. Tensor

A geometric object that generalizes scalars (rank 0), vectors (rank 1), and matrices (rank 2) to arbitrary order and transforms according to specific rules under changes of coordinate system. Second-order tensors like the stress tensor and inertia tensor are 3×3 arrays with well-defined transformation laws. Tensor analysis provides the coordinate-independent framework needed for continuum mechanics, elasticity, and general field theories.

165. Torque

The moment of a force about a point, computed as M = r × F, where r is the position vector from the reference point to the point of force application. The torque vector is perpendicular to both r and F with magnitude |r||F|sin θ. Torque represents the rotational effect of the force and is the rate of change of angular momentum in rotational dynamics.

166. Torsion

A measure of how much a space curve twists out of the osculating plane, defined as τ = −dB/ds·N, where B is the binormal and N is the principal normal. Torsion is zero for planar curves. Positive torsion means the curve twists in the direction of the right-hand rule about the tangent. Together with curvature, torsion completely characterizes a space curve up to rigid motion.

167. Triple Integral

The integral ∫∫∫_V f(x,y,z) dV of a scalar function over a three-dimensional region V. Triple integrals compute volumes (when f = 1), total masses, centers of mass, and moments of inertia of solid bodies. They are evaluated as iterated integrals, and the choice of coordinate system (Cartesian, cylindrical, or spherical) depends on the geometry of the region.

168. Triple Integral in Cylindrical Coordinates

The triple integral ∫∫∫ f(r,φ,z) r dr dφ dz, where the Jacobian factor r comes from the volume element dV = r dr dφ dz in cylindrical coordinates. Cylindrical coordinates simplify triple integrals over regions with axial symmetry, such as cylinders, cones, and paraboloids of revolution. The limits of integration must be expressed in terms of r, φ, and z.

169. Triple Integral in Spherical Coordinates

The triple integral ∫∫∫ f(r,θ,φ) r² sin θ dr dθ dφ, where the Jacobian factor r² sin θ comes from the volume element dV = r² sin θ dr dθ dφ in spherical coordinates. Spherical coordinates simplify triple integrals over regions with spherical symmetry, such as spheres, hemispheres, and spherical shells. The factor sin θ is essential and must not be omitted.

170. Unit Binormal Vector

The vector B = T × N, one of the three mutually perpendicular unit vectors in the Frenet-Serret frame. The binormal is perpendicular to the osculating plane containing T and N. It is used to describe the twisting of a space curve and appears in the formula for torsion. The rate of change of B along the curve is directly related to the torsion τ by dB/ds = −τN.

171. Unit Normal Vector

See Principal Normal Vector. More generally, a unit vector perpendicular to a curve (in 2D) or surface (in 3D) at a given point. For surface integrals, the outward unit normal n = N/|N| is used to orient the surface. The choice of unit normal determines the sign of the flux integral and must be consistent with the orientation specified in Stokes’ and Gauss’s theorems.

172. Unit Tangent Vector

The unit vector T = r'(t)/|r'(t)| pointing in the direction of increasing parameter along a curve. Since |T| = 1, the tangent vector has constant magnitude, but its direction changes along the curve. The rate of change of T with respect to arc length equals κN, where κ is the curvature and N is the principal normal, giving the first Frenet-Serret formula dT/ds = κN.

173. Unit Vector

A vector with magnitude equal to one. The unit vector in the direction of A is û = A/|A|. Unit vectors are used to specify directions without specifying magnitudes and appear in projections, normal vectors, and the Frenet-Serret frame. The Cartesian unit vectors i, j, k are the standard orthonormal basis in three dimensions and satisfy i·i = 1, i·j = 0, and i × j = k.

174. Vector

A mathematical object with both magnitude and direction, represented geometrically as an arrow in space. Vectors are defined by their magnitude and direction regardless of position (free vectors) unless specified otherwise. They satisfy the parallelogram law of addition and obey scalar multiplication. Physical quantities such as force, velocity, acceleration, displacement, and electric field are vectors.

175. Vector Addition

The operation of combining two vectors A and B to produce a resultant vector R = A + B. Geometrically, this is performed by the triangle rule (place the tail of B at the head of A) or the parallelogram rule (place both tails at a common point and draw the diagonal). Analytically, addition is performed component-wise: R = (Aₓ + Bₓ)i + (Ay + By)j + (Az + Bz)k.

176. Vector Components in Cylindrical Coordinates

The representation of a vector as F = Fᵣeᵣ + Fφeφ + Fzez in cylindrical coordinates, where eᵣ, eφ, ez are the orthonormal basis vectors. Unlike Cartesian basis vectors, eᵣ and eφ change direction as φ changes. Components in cylindrical coordinates are related to Cartesian components by rotation through angle φ. Care must be taken when differentiating vectors in cylindrical coordinates because the basis vectors are not constant.

177. Vector Components in Spherical Coordinates

The representation of a vector as F = Fᵣeᵣ + Fθeθ + Fφeφ in spherical coordinates, where eᵣ, eθ, eφ are orthonormal basis vectors that all vary in direction from point to point. The differentiation of these basis vectors introduces additional terms (Christoffel symbol contributions) that must be included when computing divergence, curl, and Laplacian in spherical coordinates.

178. Vector Field

A function that assigns a vector to every point in a region of space: F: R³ → R³. Velocity fields, force fields, electric fields, and magnetic fields are all vector fields. Vector fields are visualized using field lines (integral curves) and by arrows proportional to the field magnitude at each point. The differential operators gradient, divergence, curl, and Laplacian all operate on vector fields.

179. Vector Function

A function r(t) = x(t)i + y(t)j + z(t)k that assigns a vector to each value of the scalar parameter t. Vector functions represent the position, velocity, or any other vector quantity that varies continuously with a parameter. Calculus operations (limits, derivatives, integrals) apply to vector functions component-wise, giving the velocity r'(t), acceleration r”(t), and the arc length integral.

180. Vector Identity

One of the fundamental algebraic or differential identities satisfied by vectors and vector operators. Important identities include ∇×(∇f) = 0, ∇·(∇×F) = 0, ∇·(fF) = f∇·F + F·∇f, ∇×(fF) = f∇×F + (∇f)×F, and ∇²F = ∇(∇·F) − ∇×(∇×F). These identities are used to simplify complex vector expressions and to derive the integral theorems from the product rules.

181. Vector Multiplication

The general term covering the dot product (scalar or inner product), the cross product (vector or outer product), and the scalar triple product. There is no standard “vector division” because the product of two vectors is not always another vector. Choosing the right product for a particular physical calculation, e.g. Dot product for work, cross product for torque, is one of the useful skills gained in vector analysis.

182. Vector Operator

A differential operator that acts on scalar or vector fields to produce another field. The gradient ∇f, divergence ∇·F, curl ∇×F, and Laplacian ∇²f are the four fundamental vector operators. They appear in every PDE of engineering physics. In non-Cartesian coordinates, these operators have modified forms that include metric scale factors arising from the coordinate geometry.

183. Vector Potential

A vector function A such that F = ∇ × A for a given solenoidal (divergence-free) vector field F. The vector potential exists (by the Helmholtz theorem) whenever ∇·F = 0. In electromagnetism, the magnetic vector potential A satisfies B = ∇ × A, and the electric field in dynamic situations is E = −∇V − ∂A/∂t. The vector potential is defined up to a gauge transformation.

184. Vector Product

See Cross Product. The vector product A × B of two vectors produces a vector perpendicular to both inputs, with magnitude |A||B|sin θ. It is anti-commutative (A × B = −B × A) and is used to compute torques, angular momentum, surface normals, and the magnetic force on a moving charge F = qv × B.

185. Vector Space

A set of objects (vectors) together with operations of addition and scalar multiplication that satisfy eight axioms: commutativity and associativity of addition, existence of zero vector and additive inverses, distributivity of scalar multiplication over vector addition, distributivity over scalar addition, associativity of scalar multiplication, and the identity scalar. R³ with standard operations is the prototypical vector space.

186. Vector Subtraction

The operation A − B = A + (−B), where −B is the vector with the same magnitude as B but opposite direction. Geometrically, A − B is the vector from the tip of B to the tip of A when both are drawn from the same origin. Component-wise, A − B = (Aₓ−Bₓ)i + (Ay−By)j + (Az−Bz)k. Vector subtraction is used to compute displacement vectors and relative velocities.

187. Velocity Field

The vector field v(r, t) assigning the velocity vector to each point in a fluid domain at each instant. The velocity field is the primary unknown in fluid mechanics. Its divergence gives the volumetric strain rate (zero for incompressible flow) and its curl gives the vorticity. Line integrals of the velocity field give the circulation, and its flux through a surface gives the volumetric flow rate.

188. Velocity Vector

The derivative of the position vector with respect to time: v(t) = r'(t) = dx/dt i + dy/dt j + dz/dt k. The velocity vector is tangent to the trajectory of the particle and has magnitude equal to the speed. In curvilinear motion, the velocity is always tangential; it has no normal component. The velocity vector is the most fundamental kinematic quantity describing particle motion.

189. Volume Element

The infinitesimal volume dV used in triple integration. In Cartesian coordinates, dV = dx dy dz. In cylindrical coordinates, dV = r dr dφ dz. In spherical coordinates, dV = r² sin θ dr dθ dφ. The volume element is derived from the Jacobian determinant of the coordinate transformation. Correct use of the volume element is essential for getting correct results in triple integrals.

190. Volume Integral

The integral ∫∫∫_V f(x,y,z) dV of a scalar function over a three-dimensional region V. Volume integrals compute total mass, electric charge, thermal energy, and other distributed quantities within a volume. They are evaluated as triple iterated integrals. The Divergence theorem converts surface integrals to volume integrals and vice versa, often simplifying calculations.

191. Volume of Parallelepiped

The absolute value of the scalar triple product |A·(B × C)|, representing the volume of the three-dimensional parallelepiped with edges defined by vectors A, B, and C emanating from a common vertex. The parallelepiped volume equals the base area |B × C| times the perpendicular height |A| cos θ, where θ is the angle between A and B × C. It is zero if the three vectors are coplanar, confirming linear dependence.

192. Vortex

A region in a fluid where the flow circulates around a central axis. In potential flow theory, a point vortex has a velocity field with constant circulation and zero divergence everywhere except at the vortex center. The velocity induced by a vortex decays as 1/r. Vortices are the fundamental building blocks of two-dimensional potential flow theory and are used in the aerodynamic analysis of lifting surfaces.

193. Vorticity

The curl of the velocity field: ω = ∇ × v. Vorticity measures the local rotation rate of a fluid element; it is twice the angular velocity of that element. For irrotational (potential) flow, vorticity is zero everywhere. Vorticity is governed by the vorticity transport equation in Navier-Stokes fluid mechanics and is the key quantity in understanding turbulence, boundary layer separation, and wake dynamics.

194. Vorticity Equation

The PDE governing the evolution of vorticity in a viscous fluid, obtained by taking the curl of the Navier-Stokes equations: ∂ω/∂t + (v·∇)ω = (ω·∇)v + ν∇²ω, where ν is the kinematic viscosity. The term (ω·∇)v represents vortex stretching (absent in 2D flows) and ν∇²ω represents viscous diffusion of vorticity. The vorticity equation is a central tool in advanced fluid mechanics.

195. Wave Vector

A vector k whose direction indicates the direction of wave propagation and whose magnitude |k| = 2π/λ is the wave number. For a plane wave, the spatial variation is described by e^(ik·r). The wave vector appears in the dispersion relation ω = ω(k) and in the phase of the wave k·r − ωt. It is fundamental in acoustics, electromagnetism, and structural wave analysis.

196. Weight Function

A non-negative scalar function w(x) used to define a weighted inner product ⟨f,g⟩ = ∫ f(x)g(x)w(x)dx on a function space. Weight functions arise naturally in Sturm-Liouville problems and define the orthogonality of special function families: Legendre polynomials are orthogonal with w = 1, Laguerre polynomials with w = e^(−x), and Hermite polynomials with w = e^(−x²).

197. Work

The scalar quantity W = ∫_C F·dr = ∫_C F·T ds, measuring the total energy transferred by a force field F along a path C. For a constant force along a straight path, W = F·d where d is the displacement vector. Work is path-independent for conservative forces and path-dependent for non-conservative forces. The work-energy theorem states that the net work equals the change in kinetic energy.

198. Work Done by a Variable Force

The line integral W = ∫_C F·dr for a force field F that varies along the path C. It is computed by parametrizing the path and integrating. For a conservative force, W = V(A) − V(B), the decrease in potential energy. For non-conservative forces (friction, viscous drag), the work depends on the actual path taken and is always negative for purely resistive forces.

199. Zero Divergence Condition

The condition ∇·F = 0, which characterizes solenoidal (source-free) vector fields. The zero divergence condition ensures that the field has no net creation or annihilation of flux and all flux lines are continuous. The magnetic field always satisfies ∇·B = 0 (one of Maxwell’s equations), and the velocity field of an incompressible fluid satisfies ∇·v = 0. Solenoidal fields can always be expressed as the curl of a vector potential.

200. Zero Vector

See Null Vector. The zero vector is the unique vector of zero magnitude, serving as the additive identity in vector addition. It is the only vector with no defined direction. The gradient of a constant field is the zero vector. The cross product of parallel vectors is the zero vector. Equilibrium conditions in statics and dynamics require the resultant vector (sum of all forces and/or moments) to equal the zero vector.

[/ihc-hide-content]

201. Zonal Harmonics

Legendre polynomials Pₙ(cos θ) expressed in spherical coordinates, which are axially symmetric (independent of the azimuthal angle φ). They arise as solutions to Laplace’s equation for axially symmetric problems and are called zonal because the nodal lines of Pₙ(cos θ) divide the sphere into zones. Zonal harmonics are used in gravitational potential theory, antenna pattern analysis, and heat conduction in spherical shells.

CONCLUSION

Vector Analysis is a subject where conceptual clarity and computational skill must be developed together. For the PRC engineering board exam, the highest-priority areas are the differential operators: gradient, divergence, and curl; their physical interpretations, because these appear not just in pure mathematics problems but in electromagnetic theory, fluid mechanics, and heat transfer questions throughout all engineering disciplines. Know the Cartesian forms by heart and be comfortable with the cylindrical and spherical forms for problems involving symmetric geometries. The dot product and cross product and their applications to work, projection, torque, and angle computation are tested in almost every board exam and must be second nature.

The three great integral theorems: the Divergence theorem, Stokes’ theorem, and Green’s theorem are the most powerful tools in vector integral calculus. For the board exam, understand each theorem as a conversion rule: the Divergence theorem converts between volume integrals and closed surface integrals, Stokes’ theorem converts between open surface integrals and line integrals around the boundary, and Green’s theorem does the same in two dimensions. Know the conditions (orientability, simply connected domains, smooth boundaries) and practice identifying which theorem converts a hard integral into an easy one. These theorems appear in problems framed as electromagnetic flux, fluid flow rate, and circulation in rotating flows.

Line integrals and surface integrals, conservative fields and potential functions, and the Frenet-Serret frame for curves round out the subject. For the engineering board exam, line integrals are tested through path independence and work calculations, and the conservative field test (checking that ∇ × F = 0) is a recurring problem type. Curve geometry: curvature, torsion, unit tangent, and principal normal appears in kinematics and dynamics problems where the normal and tangential components of acceleration must be found. Build your skills in coordinate conversion (Cartesian to cylindrical to spherical) since many board problems give data in one system and require computation in another. A systematic, theorem-by-theorem review using this list as your reference will give you the coverage and depth needed to perform confidently on exam day.

For practice problems on all these topics, head over to our Electromagnetics Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.