Gradient, Divergence & Curl

Vector calculus describes how fields change in space. Learn vector-valued functions (velocity, acceleration, tangent, curvature), directional derivatives, the gradient, divergence and curl, the Laplacian, key identities, and physical meanings in heat flow, fluid flow and electromagnetism.

Heat flows down the temperature gradient; the divergence of a velocity field tells whether fluid is being created or compressed; the curl of a magnetic field is produced by current. The operator ∇ ties these ideas together and appears in Maxwell's equations and the Navier–Stokes equations.

An analogy

Gradient is the uphill arrow on a hill; divergence is how much a tap spreads water out; curl is how much a paddle wheel placed in the flow would spin.

Curves and motion

For r(t), velocity v = r′, acceleration a = r″, unit tangent T = v/|v|, curvature κ = |v × a|/|v|³ — used in road and track design and robot paths.

Gradient

Points in the direction of fastest increase; perpendicular to level surfaces. Fourier's law q = −k∇T: heat flows opposite to the gradient.

Divergence and curl

Divergence > 0: source (outflow); < 0: sink; = 0: incompressible flow. Curl measures local rotation; curl = 0 means irrotational (conservative) fields.

Physics in ∇ notation

Maxwell: ∇·E = ρ/ε₀, ∇×E = −∂B/∂t, ∇·B = 0, ∇×B = μ₀J + μ₀ε₀∂E/∂t. Heat equation ∂T/∂t = α∇²T; Laplace ∇²φ = 0 for potentials.

Formulas

• Gradient: ∇f = (f_x, f_y, f_z)
• Directional derivative: D_u f = ∇f · û (max = |∇f| along ∇f)
• Divergence: ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z
• Curl: ∇×F = (R_y − Q_z, P_z − R_x, Q_x − P_y)
• Laplacian: ∇²f = f_xx + f_yy + f_zz
• Identities: ∇×(∇f) = 0, ∇·(∇×F) = 0

Worked example: Gradient and directional derivative

Given: f = x²y + yz² at (1, 2, 1); Direction (1, 1, 0)

Find: ∇f and D_u f.

• ∇f = (2xy, x² + z², 2yz) = (4, 2, 4)
• û = (1, 1, 0)/√2
• D_u f = 6/√2 = 4.24; max rate |∇f| = 6

Answer: ∇f = (4, 2, 4); 4.24 along (1, 1, 0)

Worked example: Divergence and curl

Given: F = (x², yz, xz)

Find: ∇·F, ∇×F.

• ∇·F = 2x + z + x = 3x + z
• ∇×F = (0 − y, 0 − z, 0 − 0) = (−y, −z, 0)

Answer: 3x + z; (−y, −z, 0)

Common mistakes

• Wrong: Taking divergence of a scalar. — Right: Divergence acts on vector fields.
• Wrong: Using non-unit direction vectors. — Right: Normalise before D_u f.

Applications

• Heat transfer: q = −k∇T; heat equation.
• Fluid mechanics: Continuity ∇·v = 0; vorticity ∇×v.
• Electromagnetics: Maxwell's equations.

Key takeaways

• ∇f: direction of steepest increase.
• ∇·F: sources/sinks; ∇×F: rotation.
• ∇² appears in heat, wave, Laplace equations.
• curl grad = 0, div curl = 0.

Going further

• Curvilinear ∇: Gradient/divergence in cylindrical and spherical coordinates.
• Helmholtz decomposition: Any field = gradient + curl parts.
• Frenet–Serret frame: Tangent, normal, binormal for curves.