Vector Differentiation

 $i\times\left(\overrightarrow{\ a\ }\times i\right)+j\times\left(\overrightarrow{\ a\ }\times j\right)+k\times\left(\overrightarrow{\ a\ }\times k\right)=$  

• 0

•  $\overrightarrow{\ a\ }$  

•  $2\overrightarrow{\ a\ }$  

• 3

 $\nabla.\overrightarrow{\ r\ \ }\ =x\ \overline{i}\ +y\ \overline{i}\ +z\ \overline{k}$ ,  or  div $\overrightarrow{\ r\ }$  =

• 3

• 1

• 2

•  $\overrightarrow{\ r\ }$  

 $\nabla^2\psi\left(x,y,z\right)=0,\$  the function  $\psi$  is

• Solenoidal

• Irrotational

• Harmonic

• None of the above

 $r=\left|\overrightarrow{\ r\ }\right|,\overrightarrow{\ r\ }=xi+yj+zk,\ then\ \nabla r$  

•  $r^2$  

•  $\overrightarrow{\ r\ }$  

•  $r^3$  

•  $\frac{\overrightarrow{\ r\ }}{r}$  

 $\nabla\cdot\left(\nabla\times\overrightarrow{F}\right)$  
What will this operation result in?

• Scalar function

• Vector function

• Undefined

•  $\overrightarrow{\ F\ }$  

 $\nabla\times\left(\nabla\cdot\overrightarrow{F}\right)$  
What will this operation result in?

• Scalar function

• Vector function

• Undefined

 $\nabla\times\left(\nabla\times\overrightarrow{F}\right)=$  
What will this operation result in?

• Scalar function

• Vector function

• Undefined

• Stoke's Theorem

Find div  $\overrightarrow{F}$  :   $\overrightarrow{F}\left(x,y,z\right)=<x^2,\ y-z,\ xe^y>$  

• 2x + 1

• 2x - 1

•  $2x+xe^y$  

•  $2x+1+xe^y$  

  $\nabla\times\overrightarrow{F}$  is called 

• 0

• 1

• Divergence

• Curl

 $\overrightarrow{\ f\ }$  is irrotational if following hold

•  $\nabla\times\overrightarrow{\ f\ }=1$  

•  $\nabla.\overrightarrow{\ f\ }=0$  

•  $\nabla\times\overrightarrow{\ f\ }=0$  

•  $\nabla\cdot\overrightarrow{\ f\ }=1$