Chapter 16.R, Problem 9CC

To determine

(a)

To define:

Curl $F$

Definition:

Curl of $F:$

If $F\left(x, y, z\right)=P_{\square }\mathrm{i}+Q_{\square }\mathrm{j}+R_{\square }\mathrm{k}$ is a vector field on $R^3$ and the partial derivatives of $P, Q, R$ exists, then the curl of $F$ is vector field on $R^3$ defined by,

$\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\mathrm{ }\mathrm{F}=\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\mathrm{i}+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\mathrm{j}+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\mathrm{k}=\nabla \times \mathrm{F}$

(b)

To define:

Div $F$

Definition:

Divergence of $F:$

If $F\left(x, y, z\right)=P_{\square }\mathrm{i}+Q_{\square }\mathrm{j}+R_{\square }\mathrm{k}$ is a vector field on $R^3$ and $\frac{\partial P}{\partial x}, \frac{\partial Q}{\partial y}, \frac{\partial R}{\partial z}$ exists, then the divergence of $F$ is

$\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{F}=\mathrm{d}\mathrm{i}\mathrm{v}〈P,Q,R 〉=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}=\nabla ·F$

(c)

To explain:

The physical interpretations of curl $F$ and div $F,$ if $F$ is a velocity field in fluid flow

Explanation:

At a point in the fluid, the vector curl $F$ aligns with the axis about which the fluid tends to rotate, and its length measures the speed of rotation; div $F$ at a point measuresthe tendency of the fluidto flow away (diverge) from that point.