Curl

Relevant parts to questions...

• In suffix form: $[\nabla \times \mathbf{u}{]}_i={ϵ}_{ijk}{\partial }_j{u}_k$.
• Kill rule: $\nabla \times (\nabla f)=0$ always - because ${ϵ}_{ijk}{\partial }_j{\partial }_{k}f=0$ (antisymmetric $\times$ symmetric).
• For $\nabla \cdot (\nabla \times \mathbf{u})=0$ same trick: ${ϵ}_{ijk}{\partial }_i{\partial }_j{u}_k=0$.

The Curl of a vector field $\mathbf{u}$ is

$$\nabla \times \mathbf{u}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial }{\partial x_1}& \frac{\partial }{\partial x_2}& \frac{\partial }{\partial x_3}\\ u_1& u_2& u_3\end{array}\right|=\left(\frac{\partial u_3}{\partial x_2}-\frac{\partial u_2}{\partial x_3},\frac{\partial u_1}{\partial x_3}-\frac{\partial u_3}{\partial x_1},\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}\right)$$

In suffix notation, each component is

$$[\nabla \times \mathbf{u}{]}_i={ϵ}_{ijk}{\nabla }_j{u}_k={ϵ}_{ijk}\frac{\partial u_k}{\partial x_j}$$

Two dummy indices $j,k$ (double sum) and one free index $i$ (vector quantity).

Example

Find $\nabla \times \mathbf{A}$ at $Q=(1,2,1)$ for $\mathbf{A}=x_1^2{x}_3\mathbf{i}+2x_2\mathbf{j}+x_1{x}_2{x}_3^2\mathbf{k}$.

Component 1: ${\partial }_2{A}_3-{\partial }_3{A}_2=x_1{x}_3^2-0=x_1{x}_3^2$. Component 2: ${\partial }_3{A}_1-{\partial }_1{A}_3=x_1^2-x_2{x}_3^2$. Component 3: ${\partial }_1{A}_2-{\partial }_2{A}_1=0-0=0$.

At $Q$: $\boxed{\nabla \times \mathbf{A}=\mathbf{i}+(1-2)\mathbf{j}=\mathbf{i}-\mathbf{j}}$.

Curl of a gradient ( $\nabla \times (\nabla f)=0$)

$[\nabla \times \nabla f{]}_i={ϵ}_{ijk}{\partial }_j{\partial }_{k}f$. Mixed partials commute, so ${\partial }_j{\partial }_{k}f$ is symmetric in $j,k$. ${ϵ}_{ijk}$ is antisymmetric in $j,k$. By the kill rule, ${ϵ}_{ijk}{\partial }_j{\partial }_{k}f=0$, hence $\nabla \times \nabla f=0$.

Vector in, vector out.