Question

Let F = sin ( 8 x + 5 z ) i − 8 y e x z k . F=sin⁡(8x+5z)i−8yexzk. Calculate div ( F ) div(F) and curl ( F ) . and curl(F). (Express numbers in exact form. Use symbolic notation and fractions where needed.)

Answer 1

Required results are
`\\abla .\vec{F}`=8\cos(8x+5z)-8ye^x[/tex] and
`\\abla* \vec{F}=-8e^xz\uvec{i}+(8ye^xz+5\sin(8x+5z))\uvec{j}`

Explanation:

Given vector function is,

`\vec{F}=\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k}`

To find
`\\abla .\vec{F}` and
`\\abla * \vec{F}` .

`\\abla .\vec{F}`

`=((\partial)/(\partial x)\uvec{i}+(\partial)/(\partial y) \uvec{j}+(\partial)/(\partial z) \uvec{k})(\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k})`

`=(\partial)/(\partial x)(\sin(8x+5z))-(\partial)/(\partial z)(8ye^xz)`

`=8\cos(8x+5z)-8ye^x`

And,

`\\abla * \vec{F}`

`=((\partial)/(\partial x)\uvec{i}+(\partial)/(\partial y) \uvec{j}+(\partial)/(\partial z) \uvec{k})*(\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k})`

`\end{Vmatrix}`

`=\uvec{i}\Big[(\partial)/(\partial y)(-8ye^xz)\Big]-\uvec{j}\Big[(\partial)/(\partial x)(-8ye^xz)-(\partial)/(\partial z)(\sin(8x+5z))\Big]+\uvec{k}\Big[-(\partial)/(\partial y)(-\sin(8x+5z))\Big]`

`=-8e^xz\uvec{i}+(8ye^xz+5\sin(8x+5z))\uvec{j}`

Hence the result.