Properties of Curl and Divergence
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 6.5: “Divergence and Curl” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/6-5-divergence-and-curl |
| Textbook used in class | Stewart, Calculus, Section 16.5: “Curl and Divergence” (Vector Forms, Laplacian) |
Quick Reference
Key identities:
- $\text{curl}(\nabla f) = \mathbf{0}$ (curl of gradient is zero)
- $\text{div}(\text{curl}\,\mathbf{F}) = 0$ (divergence of curl is zero)
- $\text{div}(\nabla f) = \nabla^2 f = f_{xx}+f_{yy}+f_{zz}$ (Laplacian)
Vector forms of Green’s theorem:
- Circulation form: $\oint_C\mathbf{F}\cdot d\mathbf{r} = \iint_D(\text{curl}\,\mathbf{F})\cdot\mathbf{k}\,dA$
- Flux form: $\oint_C\mathbf{F}\cdot\mathbf{n}\,ds = \iint_D\text{div}\,\mathbf{F}\,dA$
Motivation
The del operator $\nabla$ unifies gradient, divergence, and curl:
- $\nabla f$ (applied to scalar): gradient vector
- $\nabla\cdot\mathbf{F}$ (dotted with vector): divergence (scalar)
- $\nabla\times\mathbf{F}$ (crossed with vector): curl (vector)
- $\nabla^2 f = \nabla\cdot(\nabla f)$: Laplacian (scalar)
The identities $\text{curl}(\nabla f) = \mathbf{0}$ and $\text{div}(\text{curl}\,\mathbf{F}) = 0$ correspond to the algebraic fact that $\nabla\times\nabla = \mathbf{0}$ and $\nabla\cdot(\nabla\times) = 0$ -- both follow from equality of mixed partial derivatives.
Key Concept
The two vector forms of Green’s theorem preview Stokes’ theorem and the Divergence theorem:
Circulation form says: the line integral of $\mathbf{F}$ around $\partial D$ equals the integral of $\text{curl}\,\mathbf{F}$ over $D$. (This extends to Stokes’ theorem in 3D.)
Flux form says: the outward flux of $\mathbf{F}$ through $\partial D$ equals the integral of $\text{div}\,\mathbf{F}$ over $D$. (This extends to the Divergence theorem in 3D.)
The pattern is: boundary integral = integral of a “derivative” over the enclosed region.
Worked Example
Verify $\text{curl}(\nabla f) = \mathbf{0}$ for $f(x,y,z) = x^2 yz$. (Stewart 16.5.)
$\nabla f = \langle 2xyz, x^2 z, x^2 y\rangle$.
$\text{curl}(\nabla f)$:
First component: $(x^2 y)_y - (x^2 z)_z = x^2 - x^2 = 0$.
Second component: $(2xyz)_z - (x^2 y)_x = 2xy - 2xy = 0$.
Third component: $(x^2 z)_x - (2xyz)_y = 2xz - 2xz = 0$.
$\text{curl}(\nabla f) = \mathbf{0}$. Confirmed.
$\text{curl}(\nabla f) = 0$ means $\nabla f = \mathbf{0}$. The curl of a gradient is zero, but the gradient itself is generally nonzero. The identity says the curl OPERATION on $\nabla f$ gives zero, not that $\nabla f$ itself is zero. It is analogous to the fact that $(f'(x))''\neq 0$ does not imply $f'(x) = 0$ -- these are independent properties.
Common Misconceptions
the identity $\operatorname{div}(\operatorname{curl}\mathbf{F}) = 0$ means curl fields have no effect on flux.
This is the concept-image-conflicts-definition error. The identity states that the net outward flux of $\operatorname{curl}\mathbf{F}$ through any closed surface enclosing a smooth region is zero. However, the flux of $\operatorname{curl}\mathbf{F}$ through an open surface (one with a boundary curve) can be nonzero; that nonzero value is exactly what Stokes’ theorem computes as the circulation around the boundary. The identity applies to closed surfaces; open surfaces are governed by Stokes’ theorem, not the divergence theorem.
Leveled Practice
Problem 1. For $\mathbf{F} = \nabla(x^2+y^2+z^2)$, compute $\nabla^2 f$ and verify $\text{curl}\,\mathbf{F} = \mathbf{0}$.
Show answer
$f = x^2+y^2+z^2$. $\nabla f = \langle 2x, 2y, 2z\rangle$.
$\nabla^2 f = 2+2+2 = 6$.
$\text{curl}(\nabla f) = \mathbf{0}$ by the identity (or compute directly to verify).