Recent questions tagged functions-of-two-variables

Let $C$ be the circle $(x-1)^{2}+y^{2}=25$ oriented counterclockwise. Then, the value of the line integral$$\oint_{C}\left[\left(x^{5}-3 y\right) d x+\left(-2 x+e^{y^{2}}...

Let $L$ be the lamina of the form $x^{2}+4 y^{2} \leq 64,0 \leq y \leq 4$, with density $\rho(x, y)=\mid x \mid y$. Then, the mass of $L$ (in integer) is $\_\_\_\_$

A two-dimensional source flow (with stream function, $\psi_{1}=m \tan ^{-1} \dfrac{y}{x}$ ) is placed at the origin in a uniform flow (with stream function, $\psi_{2}=U y...

​​Consider the function$$ f(x, y)=x^{2} y+2 x y^{2}-2 x^{2} y^{2}.$$Then which one of the following statements is correct?$\left(\frac{3}{2}, 0\right)$ is a point of loca...

Consider, $i$ and $j$ are unit vectors along $x$ and $y$ directions of a Cartesian $(x, y)$ coordinate system, respectively and $t$ is time. Temperature ($T$) and fluid v...

The velocity potential function in a two-dimensional flow field is given by $\phi(x, y)=-\left(a x y+b x^{2}-b y^{2}\right) \mathrm{m}^{2} / \mathrm{s}$ where $\mathrm{a}...

Given $v$ is the molar specific volume, $P$ is the pressure, $T$ is the temperature, $R$ is the Universal gas constant, and $a, b$ are van der Waal's constants.The van de...

For positive non-zero real variables $x$ and $y$, if\[\ln \left(\frac{x+y}{2}\right)=\frac{1}{2}[\ln (x)+\ln (y)]\]then, the value of $\frac{x}{y}+\frac{y}{x}$ is$1$$1 / ...

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined by $f(x, y)=\left\{\begin{array}{cc}\frac{x y}{|x|+y}, & y \neq-|x| \\ 0, & \text { otherwise. }\end{...

The surface area of the portion of the paraboloid $z=x^{2}+y^{2}$that lies between the planes $z=0$ and $z=\frac{1}{4}$ is$\frac{\pi}{6}(2 \sqrt{2}-1)$$\frac{\pi}{2}(2 \s...

Let $u(x, t)$ be the solution of the initial boundary value problem$\frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=0, \quad x \in(0,2), t>0$$u(x, 0)=...

\begin{tabular}{l|l}Q. 35 & In a steady two-dimensional compressible flow, $u$ and $v$ are the $x$ - and $y$ - \\components of flow velocity, respectively and $\rho$ is t...

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by $f(x, y)=4 x y-2 x^{2}-y^{4}+1$. The number of critical points where $f$ has local maximum is equal to