GATE EC 2013 Set 1 Q2 — Vector Analysis

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