Single-variable calculus handles functions on a line. Multivariable calculus handles functions on a plane or in space. Vector calculus goes one step further: it handles functions whose outputs are themselves vectors — quantities with both magnitude and direction, like the velocity of wind at every point in the atmosphere or the electric field surrounding a charge.
The core objects are line integrals (integrating along a path through a field), surface integrals (integrating over a surface), and the three theorems — Green's, Stokes', and the Divergence Theorem — that connect them. This page covers the essential definitions and formulas.
Vector Fields
A vector field assigns a vector to every point in space: F(x,y,z) = (P,Q,R). Examples: gravitational field, electric field, fluid velocity. Visualised as an arrow at each point.
Line Integrals
∫_C F·dr integrates a vector field along a curve C. It computes work done by force F along path C. Parametrise C as r(t), then ∫_C F·dr = ∫ₐᵇ F(r(t))·r'(t)dt.
Gradient, Divergence, Curl
- Gradient: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) — points uphill.
- Divergence: ∇·F = ∂P/∂x+∂Q/∂y+∂R/∂z — measures source/sink strength.
- Curl: ∇×F = (∂R/∂y−∂Q/∂z, ∂P/∂z−∂R/∂x, ∂Q/∂x−∂P/∂y) — measures rotation.
The Big Theorems
- Green's Theorem: relates line integrals around a closed curve to double integrals over the enclosed region.
- Stokes' Theorem: generalises Green's to surfaces in 3D — line integral around boundary = surface integral of curl.
- Divergence Theorem: surface integral of flux = triple integral of divergence. These unify all of multivariable calculus.
Vector calculus extends differentiation and integration to vector fields — functions that assign a vector to each point in space. The three key operators are gradient (∇f), divergence (∇·F), and curl (∇×F).
What is a Vector Field?
A scalar field assigns a number to each point: temperature T(x,y,z), pressure P(x,y,z). A vector field assigns a vector to each point: wind velocity at each location in the atmosphere, the electric force on a test charge at each point in space, the gravitational pull at each distance from Earth. Mathematically: F(x,y,z) = P(x,y,z)î + Q(x,y,z)ĵ + R(x,y,z)k̂. Visualised as an arrow field — one arrow per point, showing direction and magnitude.
Line Integrals — Work Along a Path
The line integral ∫_C F·dr integrates a vector field along a curve C. Physical meaning: if F is a force field, ∫_C F·dr is the work done by F on an object moving along C. Computation: parametrise C as r(t) = (x(t), y(t), z(t)) for t ∈ [a,b], then ∫_C F·dr = ∫ₐᵇ F(r(t))·r'(t)dt, which is an ordinary single integral.
Conservative Vector Fields and Potential Functions
F is conservative if F = ∇φ for some scalar potential function φ. For conservative fields, the line integral depends only on the endpoints, not the path: ∫_C F·dr = φ(B) − φ(A). This is the multivariable FTC. Test for conservativity: ∇×F = 0 (curl is zero) and the domain is simply connected. Gravity and electrostatic force are conservative. Friction and drag are not.
Green's Theorem
For a closed curve C bounding region D in 2D: ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA. This converts a line integral around the boundary into an area integral over the interior. Practical use: computing areas using line integrals (A = (1/2)∮(x dy − y dx)), and simplifying circulation calculations for fluid flow.
Stokes' and Divergence Theorems
These two theorems are the 3D generalisations of Green's Theorem. Stokes' Theorem: ∮_C F·dr = ∬_S (∇×F)·dS — the circulation around a boundary curve equals the flux of the curl through any surface bounded by that curve. Divergence Theorem: ∯_S F·dS = ∭_V (∇·F)dV — the flux out of a closed surface equals the total divergence inside the volume. Both theorems unify the fundamental theorem of calculus to higher dimensions: a boundary integral equals an interior integral of a derivative.
Surface Integrals
A surface integral ∬_S F·dS integrates a vector field F over a surface S — computing the flux of F through S. The surface element dS = n̂ dS where n̂ is the unit normal to S. Physical meaning: if F is fluid velocity, ∬_S F·dS gives the volume flow rate through S per unit time. If F is electric field E, ∬_S E·dS is the electric flux — which Gauss's Law relates to the enclosed charge.
Computing surface integrals: parametrise S as r(u,v), then dS = (∂r/∂u × ∂r/∂v) du dv, and the integral becomes ∬_D F(r(u,v))·(∂r/∂u × ∂r/∂v) du dv — an ordinary double integral.
The Unifying Picture — Generalised Stokes' Theorem
All three major theorems — the Fundamental Theorem of Calculus, Green's Theorem, Stokes' Theorem, and the Divergence Theorem — are special cases of one general statement called the Generalised Stokes' Theorem: ∫_M dω = ∫_{∂M} ω, where M is an n-dimensional manifold with boundary ∂M, ω is a differential form, and d is the exterior derivative. In this framework: the FTC is the 1D case (integrate df over [a,b] = f(b)−f(a) on {a,b}). Green's Theorem is the 2D case. Stokes' Theorem is the 3D surface case. The Divergence Theorem is the 3D volume case. Understanding this unification is the gateway to differential geometry and modern physics (general relativity, gauge theories).
Conservative Fields — Complete Characterisation
For a simply-connected domain, the following are equivalent for a C¹ vector field F: (1) F is conservative (F = ∇φ for some potential φ). (2) ∮_C F·dr = 0 for every closed curve C. (3) ∫_C F·dr depends only on the endpoints of C, not the path. (4) ∇×F = 0 (F is irrotational). Given any of these, you can find the potential φ by integration: φ(x,y) = ∫P dx (integrating the x-component with respect to x, treating y as constant) then determining the y-dependent "constant" by requiring ∂φ/∂y = Q.
- Marsden, J. & Tromba, A. (2011). Vector Calculus, 6th ed. Freeman.
- Griffiths, D.J. (2017). Introduction to Electrodynamics, Ch. 1. Cambridge UP.
- Schey, H.M. (2005). Div, Grad, Curl, and All That. Norton.