[2013EM/HMW-01]

PHY 102: Introduction to Physics-2
Tutorial-1
(Vector calculus)

1. Gradient and Divergence: If $\vec{r}$ is the position vector from a fixed point $(x_0,y_0,z_0)$ to the point $(x,y,z)$, find the following:
   • [(i)] $\vec{\nabla}\left({1\over r}\right)$.
   • [(ii)] $\vec{\nabla}(r^n)$, where $n\ne1$.
   • [(iii)] $\vec{\nabla}\cdot {\hat{r}\over r^2}$.
   where $r=|\vec{r}|$.
2. Divergence and curl of vector function $\pmb{\vec{\nu}}$:}
   • [(i)] $\vec{\nu}_1=x^2\hat{i}+3xz^2\hat{j}-2xz\hat{k}$
   • [(ii)] $\vec{\nu}_2=y^2\hat{i}+(2xy+z^2)\hat{j}+2yz\hat{k}$
3. Irrotational vector field: Find a function $g(z)$ such that the vector field ``$\vec{\nu}=2y\hat{i}(2x+g(z))\hat{j}+3yz^2\hat{k}$'' is irrotational.