Caluclus: Table of contents

Curl and Divergence


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 \(\nabla \times F\).