List of practice Questions

Consider a vector function $ \vec{F} = (y-z+2)\hat{i} + (yz+8)\hat{j} - xz\hat{k} $, on the surface (S) in the XY plane at $ z = 0 $ (as shown in the figure).
Considering Stokes theorem for space, the ABSOLUTE value of the surface integral \[ \iint_S (\nabla \times \vec{F}) \cdot \hat{n}\, dS \] is __________. (Answer in integer)

(Consider $ \hat{i}, \hat{j}, \hat{k} $, and $ \hat{n} $ as unit vectors)

Given two vectors $\vec{U}$ and $\vec{V}$, which of the following options is CORRECT for the expansion of divergence, $\nabla \cdot (\vec{U} + \vec{V})$?

Consider the function $f(x,y) = x^3 + y^2$.

If $\vec{F} = \nabla f$, then $\displaystyle\int \vec{F} \cdot d\vec{r}$ evaluated from $(0,0)$ to $(1,2)$ is ______ (answer in integer).

The gradient of a scalar field is ______.

Given a scalar field $\phi(x,y,z) = x^2 - yz$.
Magnitude of a vector in the direction of the most rapid increase of $\phi$ at a point P(3,4,1), rounded off to two decimal places, is _______________

Let $ \hat{i} $ and $ \hat{j} $ be the unit vectors in the x and y direction, respectively. The divergence $ (\nabla . \vec{F}) $ of $ \vec{F} = x^2y\hat{i} + y^3x\hat{j} $ at the point $ (3,2) $ is

Let $ \varphi $ be a scalar function. Then, $ \nabla \varphi $ is

Vector field $\vec{V}$ is defined as
\[ \vec{V} = 3x^2yz\,\hat{i} - 5xy\,\hat{j} + 6yz^2\,\hat{k} \] The curl of $\vec{V}$ at point $(2,-1,1)$ is

Let $f$ be a function of two real variables $x$ and $y$, defined as:
\[ f(x,y) = x^2 + y^2 \]
The value of the line integral $\int \vec{V}f \cdot d\vec{l}$ from point P to point Q along the path $L$ shown in the figure below is

Here $\vec{V} = \hat{x}\dfrac{\partial}{\partial x} + \hat{y}\dfrac{\partial}{\partial y}$, with $\hat{x}$ and $\hat{y}$ as the unit vectors along X and Y axes, respectively.

Curl of \[ \vec{V}=e^{xyz}(\hat{i}+\hat{j}+\hat{k}) \] at the point $(1,1,1)$ is

The divergence of \[ \left(2x^{2}z\,\hat{i}+xy^{2}z\,\hat{j}+3yz^{2}\,\hat{k}\right) \] at the point $(1,1,1)$ is

If \[ I=\int_C x^2y\,dx-xy^2\,dy, \] where $C$ is the circle \[ x^2+y^2=16, \] then $2I=$

Curl of \[ \vec{V}=e^{xyz}(\hat{i}+\hat{j}+\hat{k}) \] at the point $(1,1,1)$ is

The divergence of \[ \left(2x^{2}z\,\hat{i}+xy^{2}z\,\hat{j}+3yz^{2}\,\hat{k}\right) \] at the point $(1,1,1)$ is

If \[ I=\int_C x^2y\,dx-xy^2\,dy, \] where $C$ is the circle \[ x^2+y^2=16, \] then $2I=$

The value of $\oint_C x \, dy - y \, dx$, where $C$ is the boundary of the rectangular region formed by $x = 5$, $x = 12$, $y = 0$, and $y = 8$ is:

Let $\vec{f} = x^2 y z \hat{i} + x y^2 z \hat{j} + x y z^2 \hat{k}$ be a vector field. If $\vec{F}(x,y,z) = \text{curl }\vec{f}$, then the vector $\vec{F}(1,-2,1)$ is equal to _______}

Consider the function $ f(x,y) = (x - 2)^2(y + 3) $. Then:

The value of the improper integral $ \int_0^1 \frac{1}{4\sqrt{1-x}} dx $ is equal to _______.

The solution of \[ \frac{dy}{dx}+y=f(x), \] where \[ f(x)= \begin{cases} x, & 0\le x 0, & x\ge1, \end{cases} \] with the conditions \[ y(0)=0,\qquad y(1)=e^{-1}, \] is

If \[ \mathbf{F}=ax\mathbf{i}+by\mathbf{j}+cz\mathbf{k}, \] where $a$, $b$, and $c$ are constants, and $S$ is the surface of the unit sphere, then \[ \iint_{S}\mathbf{F}\cdot\mathbf{n}\,dS=\_ \]

If \[ \overline{a}\ \text{and}\ \overline{b} \] are constant vectors and \[ \overline{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}, \] then \[ \nabla\!\left(\overline{r}\cdot(\overline{a}\times\overline{b})\right)=\_ \]

If \[ \overline{F}=(2x^{2}-3z)\mathbf{i}-2xy\mathbf{j}-4x\mathbf{k}, \] then \[ \int_{V}\nabla\cdot\overline{F}\,dV=\_ \] where $V$ is the closed region bounded by \[ x=0,\quad y=0,\quad z=0,\quad \text{and}\quad x+2y+z=4. \]

Let S be the surface of the cube bounded by $ x = 0, x = 1, y = 0, y = 1, z = 0, z = 1 $. The value of the surface integral $ \iint_S \vec{F} \cdot d\vec{s} $ of a vector field $ \vec{F} = 4xz\hat{i} - y^2\hat{j} + yz\hat{k} $ over the entire surface S of the cube, is \dots\dots.