# Curl and Divergence

## Definition

Assume that $$\overrightarrow F:A\to\mathbb R^3$$ is a vector field, where $$A\subseteq \mathbb R^3$$. Assume that $$\overrightarrow F=\langle P,Q,R\rangle$$ where $$P$$, $$Q$$, and $$R$$ are differentiable functions on $$A$$. The curl of the vector field $$\overrightarrow F$$ is defined as $\mbox{curl }\overrightarrow F=\left\langle\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right\rangle.$ The curl of the vector field is often denoted by $$\nabla \times \overrightarrow F$$, as it can be taught of as a cross product of a formal vector $$\left\langle\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right\rangle$$ with the vector $$\overrightarrow F$$.

The divergence of the vector field $$\overrightarrow F$$ is defined as $\mbox{div }\overrightarrow F= \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}.$ The divergence is also denoted as $$\nabla \cdot \overrightarrow F$$.

Example 1.

If $$f$$ is a function and $$\overrightarrow{F}$$ a vector field defined on a domain $$D\subseteq \mathbb R^3$$ prove the following equality: $\nabla \cdot \left(f\overrightarrow F\right)=\nabla f\cdot \overrightarrow F+f\nabla\cdot \overrightarrow{F}.$

Example 2.

Assume that $$\overrightarrow F$$ is a vector field whose components have continuous second partial derivatives. Prove that $$\mbox{div }\left(\mbox{curl }\overrightarrow F\right)=0$$.

Example 3.

Assume that $$f$$ and $$g$$ are two functions that have continuous second order partial derivatives. Prove that $\nabla \cdot \left(\nabla (fg)\right)= f \nabla \cdot (\nabla g)+ g\nabla\cdot (\nabla f)+2\nabla f\cdot \nabla g.$

## Practice problems

Problem 1. Assume that the functions $$f$$ and $$g$$ are twice continuously differentiable. The expression $$\displaystyle \mbox{div }\left(\nabla f\times \nabla g\right)$$ is equal to:
• (A) $$0$$
• (B) $$\nabla \cdot \left(f\nabla g+g\nabla f\right)$$
• (C) $$\nabla \times \left(f\nabla g\right)+\nabla \times \left(g\nabla f\right)$$
• (D) $$\displaystyle 2\left(f_{xy}g_z -f_zg_{xy} - f_{yz}g_x+f_xg_{yz} \right)$$
• (E) $$\displaystyle\langle f_yg_z-f_zg_y,f_xg_z-f_zg_x,f_yg_x-f_xg_y\rangle$$

Problem 2. Find the divergence of the vector field $\overrightarrow F=x^2y^2e^z\overrightarrow i+xe^{xy}\overrightarrow j-z\cos (xy)\overrightarrow k.$

Problem 3. Find the curl of the vector field $\overrightarrow F=x^2y\overrightarrow i+ye^z\overrightarrow j-z\cos x\overrightarrow k.$

Problem 4. Find the curl of the vector field $\overrightarrow F=\langle 3x, y, z\rangle.$

Problem 5. Assume that $$\overrightarrow F=\langle z,x,2x\rangle$$. Evaluate $$\nabla \times F$$.