Recent questions in 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} \...

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] ...

​​​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 ...

​​​​​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...

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} \mat...

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$\frac{2}{3}$$\frac{4}{3}$$2$$4$

The value of the line integral $\int_{P}^{Q}\left(z^{2} d x+3 y^{2} d y+2 x z \; d z\right)$ along the straight line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is$20$$2...

Suppose that $Z \sim \mathcal{N}(0,1)$ is a Gaussian random variable with mean zero and variance $1$. Let $F(z) \equiv \mathbb{P}(Z \leq z)$ be the cumulative distributio...

Let $Q$ be a unit square in the plane with corners at $(0,0),(0,1),(1,0)$ and $(1,1)$. Let $B$ be a ball of radius $1$ in the plane centered at the origin $(0,0)$. Let $Q...

Find the vector which is closest (in Euclidean distance) to $\left(\begin{array}{lll}-1 & 1 & 1\end{array}\right)$ which can be written in the form\[a\left(\begin{array}{...

Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors such that $\vec{v}_{1}^{T} \vec{v}_{2}=0$. Let the pair of real numbers $\...

Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such ...

Given an irrotational vector field$\overrightarrow{\mathrm{F}}=(k 1 x y+k 2) \vec{a}_{x}+\left(3 x 2-k 3 Z) \vec{a}_{y}+(3 x z 2-y) \vec{a}_{z}\right.$Find $V . \vec{F}$ ...

If the electric field intensity is given by\[\mathrm{E}=\left(x u_{x}+y u_{y}+z u_{z}\right) \text { volt } / \mathrm{m} \text {, }\]the potential difference between $X(2...

$\nabla \times \nabla \times \mathrm{P},$ where $\mathrm{P}$ is a vector, is equal to$\mathrm{P} \times \nabla \times \mathrm{P}-\nabla^{2} \mathrm{P}$$\nabla^{2} \text{P...

$\displaystyle{}\iint(\nabla \times \mathrm{P}) \bullet \mathrm{ds},$ where $\mathrm{P}$ is a vector, is equal to$\displaystyle{}\oint \text{P} \bullet d l$$\displaystyle...

$ \quad f_{c} \vec{A} \cdot d \vec{t} =\int_{s} -. d \vec{s}$

The value of the integral of the function $g(x, y)=4 x^{3}+10 y^{4}$ along the straight line segment from the point $(0,0)$ to the point $(1,2)$ in the $x\text{-}y$ plane...

If the linear velocity $\overrightarrow{\mathrm{V}}$ is given by $\overrightarrow{\mathrm{V}}=x^{2} y \hat{i}+$ $x y z \hat{j}-y z^{2} k$ the anguarl velocity $\vec{w}$ a...

Given, $\mathrm{V}=x \cos ^{2} y \hat{i}+x^{2} e^{x} \hat{j}+z \sin ^{2} y \hat{k}$ and $S$ the surface of a unit cube with one corner at the origin and edges parallel to...