Let $f(x,y,z)=x^{2}yy{z}^2$ and $F=($:$yz,xz{z}^2,2yzxy)$

$($a$)$ Find gradf

$($b$)$ Compute grad$*vec(F)$

$($c$)$ Compute grad$$vec$(F)$

For $T(x,y,z)=80e^{-\frac{x^2{y}^2{z}^2}{4}}$

$($a$)$ Find gradT at $(1,1,0)$

$($b$)$ Find the directional derivative in direction vec$(v)=($:$1,2,2$:$)$

$($c$)$ In which direction does $T$ decrease most rapidly?

Prove that for any twice$-$differentiable scalar field $f,$grad$(gradf)=0$

A fluid has velocity field vec$(V)=($:$2xy,x^2-y^2,2z$:$)$

$($a$)$ Compute divergence grad$*vec(V)$

$($b$)$ Compute curl grad$$vec$(V)$

$($c$)$ Find circulation around the unit circle in the $xy-$plane