Question

Divergence of the vector field $\overrightarrow{v}(x,y,z)=(-xcosxy+y)\hat{i}+\left(ycosxy\right)\hat{j}+[\left(sin{z}^2\right)+x^2+y^2]\hat{k}$

• A. $2zcos{z}^2$
• B. $sinxy+2zcos{z}^2$
• C. $xsinxy-cosz$
• D. none of these

Answer 1

The correct option is A $2zcos{z}^2$

Let $\overrightarrow{V}=V_1\hat{i}+V_2\hat{j}+V_3\hat{k}$ then
$div\overrightarrow{V}=\frac{\partial V_1}{\partial x}+\frac{\partial V_2}{\partial y}+\frac{\partial V_3}{\partial z}$
$=\frac{\partial }{\partial x}[-xcosxy+y]+\frac{\partial }{\partial y}\left[ycosxy\right]+=\frac{\partial }{\partial z}[sin{z}^2+x^2+y^2]$
$=2zcos{z}^2$