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

Let \(S\) be the surface defined by \[ \{(x, y, z) \in \mathbb{R}^3 : z = 1 - x^2 - y^2,\, z \ge 0\}. \] Let \[ \vec{F} = -y\hat{i} + (x - 1)\hat{j} + z^2\hat{k}, \] and let \(\hat{n}\) be the continuous unit normal field to the surface \(S\) with positive \(z\)-component. Then the value of \[ \frac{1}{\pi} \iint_S (\nabla \times \vec{F}) \cdot \hat{n}\, dS \] is _________.

Let \[ \vec{F} = (y + 1)e^y \cos(x)\,\hat{i} + (y + 2)e^y \sin(x)\,\hat{j} \] be a vector field in \(\mathbb{R}^2,\) and \(C\) be a continuously differentiable path with starting point \((0,1)\) and end point \(\left(\frac{\pi}{2}, 0\right).\) Then \[ \int_C \vec{F} \cdot d\vec{r} \] equals _________.

Let \[ B = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \le 1 \} \] and define \[ u(x,y,z) = \sin\!\left(\pi(1 - x^2 - y^2 - z^2)^2\right) \] for \((x,y,z) \in B.\) Then the value of \[ \iiint_B \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \, dx\,dy\,dz \] is _________.

Consider the subset \( S = \{ (x, y) : x^2 + y^2>0 \} \) of \(\mathbb{R}^2.\) Let \[ P(x,y) = \frac{y}{x^2 + y^2}, \quad Q(x,y) = \frac{-x}{x^2 + y^2}. \] For \((x, y) \in S.\) If \(C\) denotes the unit circle traversed in the counter-clockwise direction, then the value of \[ \frac{1}{\pi} \int_C (P\,dx + Q\,dy) \] is _________.

Consider the surface \[ S = \{(x,y,xy) \in \mathbb{R}^3 : x^2 + y^2 \le 1\}. \] Let \(\vec{F} = y\hat{i} + x\hat{j} + \hat{k}\). If \(\hat{n}\) is the continuous unit normal field to the surface \(S\) with positive \(z\)-component, then \[ \iint_S \vec{F} \cdot \hat{n} \, dS \] equals

Let \( f : \mathbb{R}^3 \to \mathbb{R} \) be a scalar field, \( \vec{v} : \mathbb{R}^3 \to \mathbb{R}^3 \) be a vector field and let \( \vec{a} \in \mathbb{R}^3 \) be a constant vector. If \( \vec{r} \) represents the position vector \( x\hat{i} + y\hat{j} + z\hat{k} \), then which one of the following is FALSE?