# Questions tagged [multivariable-calculus]

Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

23,126 questions
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### Finding covariance given mean and variance of both X and Y

Say the distribution of $X$ is known, and the expected value and variance of $Y$ is known. Don't assume independence. Is this information enough to give the covariance of $X$ and $Y$? I am only ...
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### How to evaluate limits for double integral with difficult domain

Hello i am wondering what approach can be taken for evaluating a domain of a double integral given like $D=(\frac{x^2}{2} + \frac{y^2}{3})^4$ $\le \frac{xy}{\sqrt(6)}$ when i have circle or elipse i ...
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### When does a sum of two diffeomorphism still result in a diffeomorphism?

Suppose I have $f, g : \mathbb{R}^n \rightarrow \mathbb{R}^n$. Suppose $f$ and $g$ restricted to $U$ are both diffeomorphisms. I was wondering when does $f+g$ define a diffeomorphism on $U$? Is there ...
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### Problem on total derivative of bilinear form [duplicate]

Let $A$ be an invertible real $n\times n$ matrix. Define a function $F\colon\mathbb R^{n}\times \mathbb R^{n} \to\mathbb R$ by $F (x,y)=\langle Ax,y\rangle$ where $\langle x,y\rangle$ ...
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### Is this a conservative field? [on hold]

If $I=[a,b]\subseteq[0,\infty)$, $g\in C(I,\mathbb{R})$ and $f(u)=g(|u|)u$ $\forall u\in\Omega=\{y\in\mathbb{R}^n : |y|\leq 1\}$ Show that $f$ is conservative field
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### How was this integral with functions as limits solved? [duplicate]

How was this integral solved ? $$\displaystyle \frac d{dy}\int_{g(y)}^{h(y)}f(x,y)dx$$ $$=\int_{g(y)}^{h(y)}\frac \partial{\partial y} f(x,y)dx+f(h(y),y)\frac{dh(y)}{dy}-f(g(y),y)\frac{dg(y)}{dy}$$ ...
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### Generalized Laplacian?

I was wondering if any of you had ever encountered operators on $L^2(\mathbb{R}^d)$ of the form $$- \nabla \cdot A(x)\nabla$$ where $A(x)$ is some matrix field (viewed as $L^2(\mathbb{R}^{d^2}$)), ...
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### Classifying the critical points of a multivariable function

Given the function: $f(x,y)=\frac32x-\frac12x^3-xy^2$ Thanks to the gradient I managed to find that the critical points are: $(1,0),\ (-1,0),\ (0,-\sqrt{\frac32}),\ (0,+\sqrt\frac32)$ Then I found ...
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### Find $\iiint_V z$ with $V=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$

Let $f(x,y,z)=z$ and $T=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$ Find $\iiint_T f(x,y,z) dV$ I'm having a few problems with this integral, here's ...
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### Show that $f(x,y)=\frac{1}{y}$ is differentiable

Show that $f(x,y)=\frac{1}{y}$ is differentiable in its domain, i.e. $\lim_{(x,y) \rightarrow (x_0,y_0)} \frac{|\frac{1}{y} - \frac{1}{y_0} + \frac{1}{y_0^2}(y - y_0)|}{||(x,y)-(x_0,y_0)||}=0$ I ...
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### integration of a gaussian with $x^2$

I need to integrate $$\int_{-\infty}^{\infty} x^2 e^{-ax^2} \qquad \text{where } a\in R$$ The book does the following: I don't understand what's happening. I tried solving the integral using ...
1answer
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### average of sign function

Suppose we are given a unit vector $\vec{p}$, and a unit vector $\vec{\lambda}$ uniformly distributed on the hemisphere $\vec{p} \cdot \vec{\lambda} >0$. Further, let $\vec{a'}$ be a vector whose ...
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### find a limit of multivariable function [on hold]

Please ,how to prove that the limit of $f(x,y)$ when $(x,y)\to (0,0)$ is $0$ where $f(x,y)$ is $\frac{x^2}{ \sqrt{x^3 + y^3}}$ .. which is not continuous there to prove that it is continuous ...
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### Implicit differentiation applied to $z=\frac{1}{y}(f(ax+y)+g(ax-y)).$

I'm trying show that $$\frac{\partial^2z}{\partial x^2}=\frac{a^2}{y^2}\frac{\partial}{\partial y}( y^2\frac{\partial z}{\partial y})$$knowing that: $$z=\frac{1}{y}(f(ax+y)+g(ax-y)).$$ I know that,...
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### Determine the number of local maxima und minima of this function.

$f:\Bbb R^2 \to \Bbb R:x \to exp(x^2_1+x_2^2)-8x_1^2-4x_2^4$ Is there any smart way to determine the number of local maxima/minima of this function? We don't neet to find the exact points.
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### Multivariable Chain Rule - A solution I can't understand.

I am having great trouble trying to understand this chain rule question. As you can see, there are three equalities. $f(x,y) = f(w,w) = f(uv, u^2 + v^2)$ This makes absolutely no sense to me! When ...
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### Finding the extrema on sphere edges

I need to find the extrema on sphere $x^2 + y^2 = 1$ for the function $x^3 + y^3 -3xy$, i have tried to use the rail $P(t) = (\cos(t), \sin(t))$ but wasn't capable to find it's differential roots.
1answer
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### I can't find anything about this question. Please help me. Solve this problem for me

The electrical charge distribution on a circular plate of radius R meters is $\lambda(𝑟,𝜃)=𝑘\frac{𝑅}{𝑟}\sin(\frac{\pi}{𝑅}𝑟)(1−\sin\frac{𝜃}{2})\frac{C}{m^2}$ is given ($k$ is a constant). ...
1answer
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### Multivariable Chain Rule - How to solve this?

$g(x,y) = f(x^2 - y^2, 2xy)$ How do I find the the partial derivative of $g$ with respect to $x$ (in terms of $f$) in this case? Thanks for the help! I honestly can't figure out how to do this :(
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### Discuss the function for continuity at (0,0) [on hold]

$$f(x,y)=\begin{cases} 0,& (x,y)=(2y,y)\\ \exp[|x-2y|/(x^2-4xy+4y^2)],& (x,y)≠(2y,y).\end{cases}$$ Please give full answer to help me understand .