List of top Multivariable Calculus Questions

Let \( D = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \le 1\} \) and \( f : D \to \mathbb{R} \) be a non-constant continuous function. Which of the following is TRUE?

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}, \, x, y, z \in \mathbb{R}. \) Which of the following is FALSE?

Let \( S \) be the part of the cone \( z^2 = x^2 + y^2 \) between the planes \( z = 0 \) and \( z = 1. \) Then the value of the surface integral \( \iint_S (x^2 + y^2) \, dS \) is

Let \( \vec{F} = x\hat{i} + y\hat{j} + z\hat{k} \) and \( S \) be the sphere given by \( (x - 2)^2 + (y - 2)^2 + (z - 2)^2 = 4. \) If \( \hat{n} \) is the unit outward normal to \( S \), then \( \dfrac{1}{\pi} \iint_S \vec{F} \cdot \hat{n} \, dS \) is .............

Let \( \varphi(x, y, z) = 3y^2 + 3xyz \) for \( (x, y, z) \in \mathbb{R}^3 \). Then the absolute value of the directional derivative of \( \varphi \) in the direction of the line \[ \frac{x-1}{2} = \frac{y-2}{-1} = \frac{z}{-2} \] at the point \( (1, -2, 1) \) is .............

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be given by

\[ f(x, y) = \begin{cases} \frac{x^2 y (x - y)}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} \]

Then

\[ \frac{\partial}{\partial x} \left( \frac{\partial}{\partial y} f \right) - \frac{\partial}{\partial y} \left( \frac{\partial}{\partial x} f \right) \text{ at the point } (0, 0) \text{ is ..........} \]

Let \( f(x, y) = \sqrt{3 x^3 y \sin \left( \frac{\pi}{2} e^{(x - 1)} \right) + xy \cos \left( \frac{\pi}{3} e^{(y - 1)} \right)} \) for \( (x, y) \in \mathbb{R}^2, x > 0, y > 0 \).
Then \( f_x(1, 1) + f_y(1, 1) = \) ..........

The area of the parametrized surface \[ S = \left\{ \left( (2 + \cos u) \cos v, (2 + \cos u) \sin v, \sin u \right) \in \mathbb{R}^3 \mid 0 \leq u \leq \frac{\pi}{2}, 0 \leq v \leq \frac{\pi}{2} \right\} \] is .................. (correct up to two decimal places).

If the volume of the solid in \( \mathbb{R}^3 \) bounded by the surfaces \[ x = -1, \ x = 1, \ y = -1, \ y = 1, \ z = 2, \ y^2 + z^2 = 2 \] is \( \alpha - \pi \), then \[ \alpha = \text{.........}. \]

The value of the integral \[ \int_0^1 \int_0^1 y e^{x y^2} \, dy \, dx \text{ is ............} \text{ (correct up to three decimal places)}. \]

Consider the region \( D \) in the yz-plane bounded by the line \( y = \frac{1}{2} \) and the curve \( y^2 + z^2 = 1 \), where \( y \geq 0 \). If the region \( D \) is revolved about the \( z \)-axis in \( \mathbb{R}^3 \), then the volume of the resulting solid is

If \( \hat{F}(x, y) = (3x - 8y) \hat{i + (4y - 6xy) \hat{j} \) for \( (x, y) \in \mathbb{R}^2 \), then \( \oint_C \hat{F} \cdot d \hat{r} \), where \( C \) is the boundary of the triangular region bounded by the lines \( x = 0 \), \( y = 0 \), and \( x + y = 1 \) oriented in the anti-clockwise direction, is}

Let \( f(x, y) = \begin{cases \frac{xy}{(x^2 + y^2)^{\alpha}}, & (x, y) \neq (0,0)
0, & (x, y) = (0,0) \end{cases} \) Then which one of the following is TRUE for \( f \) at the point \( (0, 0) \)?}

Let \( a, b \in \mathbb{R} \) and let \( f: \mathbb{R} \to \mathbb{R} \) be a thrice differentiable function. If \( z = e^u f(v) \), where \( u = ax + by \) and \( v = ax - by \), then which one of the following is TRUE?

The tangent plane to the surface \( z = \sqrt{x^2 + 3y^2} \) at (1, 1, 2) is given by

In \( \mathbb{R}^3 \), the cosine of the acute angle between the surfaces \( x^2 + y^2 + z^2 - 9 = 0 \) and \( z - x^2 - y^2 + 3 = 0 \) at the point (2, 1, 2) is