Vector Calculus
Vector calculus is the calculus of vector-valued functions in two and three dimensions. It provides the language of fluid mechanics, electromagnetism and continuum physics, and its four central theorems — the fundamental theorem for line integrals, Green's theorem, Stokes' theorem and the divergence theorem — generalise the Fundamental Theorem of Calculus to higher dimensions.
A function that assigns a vector to every point in a region of space, e.g., F(x, y, z) = (P, Q, R). Physical examples include velocity fields of fluids, gravitational fields, and electromagnetic fields.
The del operator
The vector differential operator ∇ ("del" or "nabla") is defined in Cartesian coordinates as
∇ = (∂/∂x, ∂/∂y, ∂/∂z)
It generates three fundamental field operations:
| Operation | Symbol | Acts on | Produces | Geometric meaning |
|---|---|---|---|---|
| Gradient | ∇f | scalar f | vector | Direction of steepest increase |
| Divergence | ∇·F | vector F | scalar | Net outflow per unit volume |
| Curl | ∇×F | vector F | vector | Local rotation (circulation per area) |
Two important second-derivative identities:
- curl(grad f) = 0 for any twice-differentiable scalar field.
- div(curl F) = 0 for any twice-differentiable vector field.
The Laplacian is ∇²f = ∇·(∇f) = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z². It governs steady-state diffusion, electrostatic potentials and wave propagation.
Line integrals
The line integral of a scalar field f along a curve C parameterised by r(t), t ∈ [a, b] is
∫_C f ds = ∫_a^b f(r(t)) |r'(t)| dt
For a vector field F:
∫_C F · dr = ∫_a^b F(r(t)) · r'(t) dt
This represents the work done by F along C. If F = ∇φ (F is conservative), the line integral depends only on endpoints:
∫_C ∇φ · dr = φ(end) − φ(start)
This is the fundamental theorem for line integrals.
Surface integrals
The flux of a vector field F through an oriented surface S is
∬_S F · dA = ∬_S F · n̂ dS
where n̂ is the unit normal. In Cartesian form with dA = (dy dz, dz dx, dx dy), it measures the volumetric flow of F through S per unit time.
- A vector field F is conservative iff ∇×F = 0 on a simply connected domain; equivalently F = ∇φ for some scalar potential φ.
- A vector field is solenoidal (incompressible) iff ∇·F = 0; locally F = ∇×A for some vector potential A.
- An irrotational field has zero curl; a divergence-free field has zero divergence.
- Electromagnetic field decomposition (Helmholtz theorem): any well-behaved vector field is the sum of a gradient and a curl.
The integral theorems
These four theorems link integrals over a region to integrals over its boundary — generalisations of ∫_a^b f'(x) dx = f(b) − f(a).
Green's theorem (plane)
For a positively oriented closed curve C bounding a region D ⊂ R²:
∮_C P dx + Q dy = ∬_D (∂Q/∂x − ∂P/∂y) dA
It converts a line integral around a planar loop into a double integral over the enclosed area — useful for computing areas (set P = 0, Q = x) and for proving conservation in 2-D fluid flow.
Stokes' theorem
For a smooth oriented surface S with boundary curve ∂S:
∮_∂S F · dr = ∬_S (∇×F) · dA
It generalises Green's theorem to a 2-D surface in 3-D space. In electromagnetism it is the integral form of Faraday's law.
Divergence theorem (Gauss)
For a region V with closed boundary surface ∂V (outward normal):
∯_∂V F · dA = ∭_V (∇·F) dV
Volume integral of divergence = total flux out of the bounding surface. Underlies Gauss's law for electric fields (∮ E · dA = Q/ε₀), Archimedes' principle, and the continuity equation of fluid flow.
A useful mnemonic for the three integral theorems: 1-D: f(b) − f(a) = ∫f' dx 2-D: boundary line integral = area integral of (Q_x − P_y) 3-D surface: boundary loop = surface integral of curl 3-D volume: boundary surface = volume integral of divergence Each step generalises "integral of derivative = boundary value" to one higher dimension.
Coordinate systems
Most physical problems exploit symmetry through alternative coordinate systems.
Cylindrical (r, φ, z)
Volume element dV = r dr dφ dz; ∇·F = (1/r) ∂(rF_r)/∂r + (1/r) ∂F_φ/∂φ + ∂F_z/∂z.
Spherical (r, θ, φ)
Volume element dV = r² sin θ dr dθ dφ; the Laplacian of a radially symmetric function f(r) is ∇²f = (1/r²) d(r² df/dr)/dr.
The right choice of coordinates can reduce a difficult 3-D problem to a 1-D integration — essential for problems with spherical or axial symmetry.
Applications
- Electromagnetism: Maxwell's equations are written compactly in terms of ∇·E, ∇·B, ∇×E, ∇×B.
- Fluid mechanics: continuity (∂ρ/∂t + ∇·(ρv) = 0), Navier–Stokes, vorticity (ω = ∇×v).
- Heat and diffusion: temperature satisfies ∂T/∂t = α∇²T (Fourier's law combined with energy balance).
- Gravitation: Poisson's equation ∇²φ = 4πGρ.
These applications make vector calculus indispensable for FPSC Applied Mathematics, civil/mechanical engineering, geophysics and theoretical physics.