Problem $5\mathrm{.}31$

$($a$)$ Complete the proof of Theorem $2,$ Sect. $1\mathrm{.}6\mathrm{.}2.$ That is$,$ show that any diver$-$

genceless vector field F can be written as the curl of a vector potential A$.$ What

you have to do is find Ax $,$ Ay $,$ and Az such that $($i$)$ Az $/$y$$ Ay $/$z $=$ Fx ;

$($ii$)$ Ax $/$z$$ Az $/$x $=$ Fy ; and $($iii$)$ Ay $/$x$$ Ax $/$y$=$ Fz $.$ Here$$s one

way to do it: Pick Ax $=0,$ and solve $($ii$)$ and $($iii$)$ for Ay and Az $.$ Note that

the $$constants of integration$$ are themselves functions of y and z$$they$$re

constant only with respect to x$.$ Now plug these expressions into $($i$),$ and use

the fact that $$ F$=0$ to obtain

Ay $=$

x

Fz $($x$$

$0$

$,$ y$,$ z$)$ d x$$; Az $=$

y

Fx $(0,$ y$$

$,$ z$)$ d y$$

$0$

$$

$0$

x

Fy $($x$$

$,$ y$,$ z$)$ d x$$

$.$

$($b$)$ By direct differentiation, check that the A you obtained in part $($a$)$ satisfies

$$ A$=$ F$.$ Is A divergenceless? $[$This was a very asymmetrical construc$-$

tion, and it would be surprising if it were$$although we know that there exists

a vector whose curl is F and whose divergence is zero.$]$

$($c$)$ As an example, let F$=$ y$$

x $+$ z$$

y $+$ x$$

z$.$ Calculate A$,$ and confirm that

$$ A$=$ F$.($For further discussion, see Prob. $5\mathrm{.}53.)$