Curl of a vector field

In the plane

Following @needham2021visual page 262, we define first the circulation of a vector field $V$ along a closed simple loop $L$ :

C_{L} (V) \equiv \oint_{L} V \cdot d r = \oint_{L} V_{L} d s = \oint_{L} [P d u + Q d v]

Then, it can be shown that

C_{L} (V) ≍ {Q (c) - Q (a)} δ v - {P (d) - P (b)} δ u ≍

≍ (\partial_{u} Q - \partial_{v} P) δ A

where $≍$ means "ultimately equal".
Now, the curl is defined as the "local circulation per unit area", so

c u r l (V) = \partial_{u} Q - \partial_{v} P

which is the traditional definition.

In the 3D space $R^{3}$

Mnemonic rule

\nabla \times F = | \begin{array}{ccc} \hat{ı} & \hat{ȷ} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{array} |

which expands as

\nabla \times F = (\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}) \hat{i} + (\frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x}) \hat{j} + (\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}) \hat{k} = [\begin{array}{l} \frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z} \\ \frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x} \\ \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y} \end{array}]

In $R^{3}$ , for vector fields $F = (F^{1}, F^{2}, F^{3})$ , the curl is alternatively given by:

(\nabla \times F)^{i} = ϵ_{k}^{i j} \partial_{j} F^{k},

where $ϵ_{k}^{i j}$ is the Levi-Civita symbol, and the Einstein summation convention is used.

In a coordinate free manner (provided that we have a metric, like the standard one), the curl can be defined as

(* (d F^{♭}))^{♯},

where we are using the musical isomorphism $♭, ♯$ and the Hodge star operator $*$ .

Transformation of the curl

Given a vector field $V (p)$ and a linear map $A (p)$ applied in every tangent space $T_{p} M$ , we can find an expression for the curl of the vector field $A (p) V (p)$ . Define the vector field $W (p) = A (p) V (p)$ .

Given $W = A V$ , we can expand:

W^{k} = A_{m}^{k} V^{m} .

Let's compute $(\nabla \times W)_{i}$ :

(\nabla \times W)^{i} = ϵ_{k}^{i j} \partial_{j} (A_{m}^{k} V^{m}) .

Using the product rule:

(\nabla \times W)^{i} = ϵ_{k}^{i j} \partial_{j} (A_{m}^{k}) V^{m} + ϵ_{k}^{i j} A_{m}^{k} \partial_{j} (V^{m}) .

Special Case: $A$ is Independent of $p$
If $A$ does not depend on $p$ (i.e., $A (p) = A$ is a constant linear map across $M$ ), then $\partial_{j} (A_{m}^{k}) = 0$ , and the expression simplifies to:

(\nabla \times W)^{i} = ϵ_{k}^{i j} A_{m}^{k} \partial_{j} (V^{m}),

or simply

\nabla \times (A V) = A (\nabla \times V) .

Curl of a vector field

In the plane

In the 3D space R3

Transformation of the curl

In the 3D space $R^{3}$