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

Consider a volume V enclosed by a closed surface S having unit surface normal \(\hat{n}\). For \(\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}\), the value of the surface integral \(\frac{1}{9} \oint_{S} \mathbf{r} \cdot \hat{n} \,dS\) is

In Cartesian coordinates, consider the functions \(u(x, y) = \frac{1{2}(x^2 - y^2)\) and \(v(x,y) = xy\). If \((r, \theta)\) are the polar coordinates, the Jacobian determinant \(|\frac{\partial(u,v)}{\partial(r,\theta)}|\) is}

Given a function \(f(x, y) = \frac{x}{a}e^y + \frac{y}{b}e^x\), where \(x = at\) and \(y = bt\) (a and b are non-zero constants), the value of \(\frac{df}{dt}\) at \(t = 0\) is

Which one of the following figures represents the vector field \(\mathbf{A} = y\hat{i}\)?
(\(\hat{i}\) is the unit vector along the x-direction)

The gradient of a scalar field \( S(x, y, z) \) has the following characteristic(s).

If \( \phi(x,y,z) \) is a scalar function which satisfies the Laplace equation, then the gradient of \( \phi \) is