List of top Physics Questions on Vector Calculus asked in IIT JAM PH

Consider a vector $\mathbf{F} = \frac{1}{\pi}[-\sin y \hat{i} + x(1 - \cos y)\hat{j}]$. The value of the integral $\oint \mathbf{F} \cdot d\mathbf{r}$ over a circle $x^2 + y^2 = 1$ evaluated in the anti-clockwise direction is _____. (in integer)

Consider a vector function $\vec{u}(\vec{r})$ and two scalar functions $\psi(\vec{r})$ and $\phi(\vec{r})$. The unit vector $\hat{n}$ is normal to the elementary surface $dS$, $dV$ is an infinitesimal volume, $d\vec{l}$ is an infinitesimal line element, and $\partial / \partial n$ denotes the partial derivative along $\hat{n}$. Which of the following identities is/are correct?

The line integral of the vector function $u(x, y) = 2y \, \hat{i} + x \, \hat{j}$ along the straight line from (0, 0) to (2, 4) is ..........

The volume integral $\displaystyle \int_V e^{-\left(\frac{r}{R}\right)^2} \vec{\nabla} \cdot \left(\frac{\hat{r}}{r^2}\right) d^3r$, where $V$ is the volume of a sphere of radius $R$ centered at the origin, is equal to:

A curve is given by $\vec{r}(t)=t\hat{i}+t^{2}\hat{j}+t^{3}\hat{k}$. The unit vector of the tangent at $t=1$ is

Let $f(x,y)=x^{3}-2y^{3}$. The curve along which $\nabla^{2} f = 0$ is