GATE EC Vector Analysis questions

A surface is given by $z^{2}=2 x^{2}-y^{2}$ and $\vec{n}$ and $-\vec{n}$ are unit normal vectors to the surface at the point $\vec{P}=\hat{\boldsymbol{\imath}}+\sqrt{2} \widehat{\boldsymbol{k}}$. Which of the following vectors can be $\vec{n}$, where $\hat{\boldsymbol{\imath}}, \hat{\boldsymbol{\jmath}}$ and $\widehat{\boldsymbol{k}}$ are the …

Consider the vectors $$\boldsymbol{a}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \quad \boldsymbol{b}=\left[\begin{array}{c} 0 \\ 3 \sqrt{2} \end{array}\right]$$ For real-valued scalar variable $x$, the value of $$\min _{x}\|\boldsymbol{a} x-\boldsymbol{b}\|_{2}$$ is $\_\_\_\_\_$ (rounded off to two decimal places). $\|\cdot\|_{2}$ denotes the …

​​​Let $\hat{i}$ and $\hat{j}$ be the unit vectors along $x$ and $y$ axes, respectively and let $A$ be a positive constant. Which one of the following statements is true for the vector fields $\overrightarrow{F_{1}}=A(\hat{i} y+\hat{j} x)$ and $\overrightarrow{F_{2}}=A(\hat{i} y-\hat{j} x) ?$ .

​​​​​Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho}{\partial t}$ is continuous. Let $V$ be an arbitrary volume in space enclosed by the closed surface $S$ and $\hat{n}$ be the outward unit normal of $S$ Which of the following …

Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y+F_{3} \mathrm{~d} z\right)$ evaluates to the same value along any path $C$ that starts at $A$ and ends at $B$. Then which of the following …

The rate of increase, of a scalar field $f(x, y, z)=x y z$, in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is

Consider the two-dimensional vector field $\overrightarrow{\rm F}(x, y) = x \overrightarrow{i} + y \overrightarrow{j},$ where $\overrightarrow{i}$ and $\overrightarrow{j}$ denote the unit vectors along the $x –$axis and the $y –$axis, respectively. A contour $\text{C}$ in the $x – y$ plane, as shown in the figure, is composed of two horizontal …

In an electrostatic field, the electric displacement density vector, $\overrightarrow{D},$ is given by $$\overrightarrow{D} (x, y, z) = (x^{3} \overrightarrow{i} + y^{3} \overrightarrow{j} + xy^{2} \overrightarrow{k}) \; \text{C/m}^{2},$$ where $\overrightarrow{i}, \overrightarrow{j}, \overrightarrow{k}$ are the unit vectors along …

Consider the vector field $F\:=\:a_{x}\left ( 4y-c_{1}z \right )+a_y\left ( 4x + 2z\right )+a_{z}\left ( 2y +z\right )$ in a rectangular coordinate system $(x,y,z)$ with unit vectors $a_{x},\:a_{y}$ and $a_{z}$. If the field $F$ is irrotational (conservative), then the constant $c_{1}$ (in integer) is _________________

For a vector field $D=\rho\cos^{2}\:\varphi \:a_{\rho }+z^{2}\sin^{2}\:\varphi \:a_{\varphi }$ in a cylindrical coordinate system $\left ( \rho ,\varphi ,z \right )$ with unit vectors $a_{\rho },a_{\varphi }$ and $a_{z}$, the net flux of $D$ leaving the closed surface of the cylinder $\left ( \rho =3, 0\leq z\leq 2 \right )$ (rounded off to two …

For a vector field $\overrightarrow{A}$, which one of the following is $\text{FALSE}$?

Consider the line integral $$\int_{c} (xdy-ydx)$$ the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2 \times 3$ rectangle and a semi-circle of radius $1$. The line integral evaluates to

The smaller angle (in degrees) between the planes $x+y+z=1$ and $2x-y+2z=0$ is ________.

If the vector function $\overrightarrow{F}=\widehat{a_x}(3y-k_1z)+\widehat{a_y}(k_2x-2z)-\widehat{a_z}(k_3y+z)$ is irrotational, then the values of the constants $k_1$,$k_2$ and $k_3$, respectively, are

Consider the time-varying vector $\textbf{I}=\hat{x}15\cos(\omega t)+\hat{y}5\sin(\omega t)$ in Cartesian coordinates, where $\omega> 0$ is a constant. When the vector magnitude $\mid \textbf{I} \mid$ is at its minimum value, the angle $\theta$ that $\textbf{I}$ makes with the $x$ axis (in degrees, such that $0\leq \theta \leq 180)$ ________

Suppose $C$ is the closed curve defined as the circle $x^{2}+y^{2}=1$ with $C$ oreinted anti-clockwise. The value of $\oint$ ( $xy^{2}$ $dx$ + $x^{2}y$ $dy$ )over the curve $C$ equals _________

A vector $\overrightarrow{P}$ is given by $\overrightarrow{P} = x^3y\overrightarrow{a}_x – x^2y^2\overrightarrow{a}_y – x^2 yz \overrightarrow{a}_z$. Which one of the statements is TRUE?

The electric field intensity of a plane wave traveling in free space is given by the following expression $$\textbf{E}(x,t)=\textbf{a}_y \: 24 \: \pi \: \: \cos(\omega t – k_0 x) \: \: (V/m)$$ In this field, consider a square area $10 \: \text{cm} \times 10 \: \text{cm}$ on a plane $x+y=1$. The total time-averaged power (in mW) passing through the square …

Consider a uniform plane wave with amplitude $(E_0)$ of $10 \: V/m$ and $1.1$ GHz frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity $(\varepsilon _r)$ and permeability $(\mu _r)$ as shown in the figure. The magnitude of the transmitted electric field component (in V/m) after it has …

A vector field $\textbf{D} = 2\rho^{2}\:\textbf{a}_{\rho} + z\: \textbf{a}_{z}$ exists inside a cylindrical region enclosed by the surfaces $\rho =1,z = 0$ and $z = 5.$ Let $S$ be the surface bounding this cylindrical region. The surface integral of this field on $S(∯_{S} \textbf{D.ds})$ is _______.