16.5 Videos Guide

16.5a

• The del operator

  • $\mathbf{\nabla}=\left\langle \frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{\partial}{\partial z}\right\rangle =\frac{\partial}{\partial x}\mathbf{i}+\ \frac{\partial}{\partial y}\mathbf{j}+\ \frac{\partial}{\partial z}\mathbf{k}$

• The curl of a vector field $\mathbf{F}(x,\ y,\ z)=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$

  • $curl\mathbf{\ F}=\mathbf{\nabla}\times\mathbf{F}=\left|\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ P & Q & R \end{matrix}\right|$

Theorem (statement and proof):

• If $\mathbf{F}$ is conservative, then $\mbox{curl}\mathbf{\ F}=\mathbf{0}$ (so if $\mbox{curl}\ \mathbf{F}\neq\mathbf{0}$, then $\mathbf{F}$ is not conservative)

Theorem (statement):

• The converse of the above theorem is true if $\mathbf{F}$ is defined on all of $\mathbb{R}^{3}$

16.5b

• The divergence of a vector field $\mathbf{F}(x,\ y,\ z)=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$

  • $\mbox{div}\ \mathbf{F}=\mathbf{\nabla}\cdot\mathbf{F}=\left\langle \frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{d}{dz}\right\rangle \cdot\left\langle P,\ Q,\ R\right\rangle =\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$

Exercise:

• Find (a) the curl and (b) the divergence of the vector field.
  $$\mathbf{F}(x,\ y,\ z)=x^{3}yz^{2}\ \mathbf{j}+y^{4}z^{3}\ \mathbf{k}$$

• Zero results

  • If $\mbox{curl}\ \mathbf{F}=\mathbf{0}$ at $P$, then $\mathbf{F}$ is said to be irrotational at $P$.

  • If $\mbox{div}\ \mathbf{F}=0$, then $\mathbf{F}$ is said to be incompressible.

Theorem (statement and proof):

• If $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ and $P$, $Q$, and $R$ have continuous second-order partial derivatives, then $\mbox{div}\ \mbox{curl}\ \mathbf{F}=\mathbf{0}$.

16.5c

• Vector forms of Green’s Theorem

  • $\oint^{\ }_{C}{\mathbf{F}\cdot d\mathbf{r}=\iint^{\ }_{D}{\left(\mbox{curl}\ \mathbf{F}\right)\cdot\mathbf{k}\ dA}}$

  • $\oint^{\ }_{C}{\mathbf{F}\cdot\mathbf{n\ }ds}=\iint^{\ }_{D}{\mbox{div}\ \mathbf{F}(x,\ y)\ dA}$