Respuesta :
Answer:
Required results are [tex]\nabla .\vec{F}[/tex]=8\cos(8x+5z)-8ye^x[/tex] and [tex]\nabla\times \vec{F}=-8e^xz\uvec{i}+(8ye^xz+5\sin(8x+5z))\uvec{j}[/tex]
Step-by-step explanation:
Given vector function is,
[tex]\vec{F}=\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k}[/tex]
To find [tex]\nabla .\vec{F}[/tex] and [tex]\nabla \times \vec{F}[/tex] .
[tex]\nabla .\vec{F}[/tex]
[tex]=(\frac{\partial}{\partial x}\uvec{i}+\frac{\partial}{\partial y} \uvec{j}+\frac{\partial}{\partial z} \uvec{k})(\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k})[/tex]
[tex]=\frac{\partial}{\partial x}(\sin(8x+5z))-\frac{\partial}{\partial z}(8ye^xz)[/tex]
[tex]=8\cos(8x+5z)-8ye^x[/tex]
And,
[tex]\nabla \times \vec{F}[/tex]
[tex]=(\frac{\partial}{\partial x}\uvec{i}+\frac{\partial}{\partial y} \uvec{j}+\frac{\partial}{\partial z} \uvec{k})\times(\sin(8x+5z)\uvec{i}-8ye^xz\uvec{k})[/tex]
[tex]\end{Vmatrix}[/tex]
[tex]=\uvec{i}\Big[\frac{\partial}{\partial y}(-8ye^xz)\Big]-\uvec{j}\Big[\frac{\partial}{\partial x}(-8ye^xz)-\frac{\partial}{\partial z}(\sin(8x+5z))\Big]+\uvec{k}\Big[-\frac{\partial}{\partial y}(-\sin(8x+5z))\Big][/tex]
[tex]=-8e^xz\uvec{i}+(8ye^xz+5\sin(8x+5z))\uvec{j}[/tex]
Hence the result.
