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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).

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16 views

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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0answers
12 views

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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0answers
23 views

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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2answers
33 views

Why do we choose the limits for double integrals in this way?

This is a general question which I have been asking (to no avail) for a while around my class. I will ask my question using an example. Evaluate $I=\displaystyle\int_{0}^{1}\mathrm{d}x\displaystyle\...
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0answers
3 views

How do you find the gradient of the edges (error estimation) in finite elements?

I am using the finite element method and need to find the errors associated with each of my elements. I am looking for help to find the error on the edges of the triangle, preferably by hand and not ...
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1answer
28 views

Vector Calculus integration region.

Evaluate $ \int_0^\sqrt2\int_0^{3y}\int_{x^2+3y^2}^{8-x^2-y^2}dzdxdy$. The method given in the answer booklet was to calculate the integrals one at a time and get a numerical answer and it is quite ...
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1answer
17 views

Integrating double integral with spherical coordinates problem with interpret a domain

Hello i have the following problem i am solving integral with spherical coordinates but i am getting wrong answer - i think i am integrating correct so i think the problem is coming from the ...
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0answers
44 views

finding limit of a multivariable function

I want to find the limit of $\dfrac{x^2 +y^4}{y^2+x^4}$ as $(x,y) \to (\infty, \infty)$. I tried two different paths: $y=x$: We have the limit of the above function $\lim (x^2+x^4)/(x^4+x^2)=1$, $...
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2answers
28 views

Can we use the chain rule here? Is that a parametric function? [on hold]

So I have this problem For the function $f(u,v)$ find $f'_x , f'_y$ if $u = x^2y^2 $, $v = xy$ So.I thought that I can find $f'_x , f'_t$ by just computing partial derivative for both For $u = x^2y^...
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1answer
20 views

Surface area inside cylinder

Find the surface area of the part $\sigma$: $x^2+y^2+z^2=4$ that lies inside the cylinder $x^2+y^2=2y$ So, the surface is a sphere of $R=2$. It looks there should be double integral to calculate the ...
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1answer
28 views

Is there a way to mathematically prove $\psi (\mathbf{r})$ varies continuously (using the intuitive arguments provided below)?

Electric potential at a point outside the charge distribution is: $\displaystyle \psi (\mathbf{r})= \int_{V'} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ where: $\mathbf{r}=(x,y,z)$ ...
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2answers
13 views

Parameterize union of two shapes in 3space

I am supposed to parameterize the union of the two shapes $x^2 + y^2 = 1, z = y$. I do not even know how to get the union of the two shapes. When I graph the two shapes the intersection does not ...
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0answers
16 views

Limit of multivariate fraction in the shape of Young's Inequality

We are asked to determine the conditions s.t the following limit exists. We need to determine an inequality with a,b,c,d s.t. this limit exists. However, I am unsure on how to proceed. I thought about ...
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2answers
28 views

Physical Meaning of Volume (or surface) Integrals with limits

So when we do triple integrals of a 3D object, and apply the own object as the limit, we will get the volume of the object (from slicing the object and add them together) But if we apply a different ...
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1answer
29 views

Matrix trace gradient

Given $A, B$ positive definite matrices and $f(M) = \textrm{tr}(A^{-1}M)+\textrm{tr}(M^{-1}B)$, what is $\nabla f(N)$ ? According to my source, $\nabla f(N)=A-N^{-1}BN^{-1}$. However, I would expect ...
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1answer
20 views

Example of Integration by Parts in Higher Dimension

I'm looking for a concrete example of an application of integration by parts in higher dimensions. The formula I'm looking at is from here, here, and here. $\Omega$ is an open bounded subset of $\...
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2answers
39 views

find limit of a multivariable function

I have to show continuity at $(0,0)$ of $f(x,y)=\frac{\sin(x^2) + \sin(y^2)}{\sqrt{x^2 +y^2}}$ for $(x,y)\ne(0,0)$ and $f(0,0)=0$. I tried to find the limit using polar coordinates $ \frac{\sin(r^2\...
2
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1answer
34 views

Using $|r|$ versus using $r$ for double integration

Suppose that I have a double integral where the integrand has variables $x$ and $y$, and I am using the polar substitution $x=r\cos(\theta)$ and $y=r\sin(\theta)$. Suppose the region of integration is ...
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0answers
26 views

This paper implies that $a \frac{\partial{b^\ast}}{\partial{q}} = b \frac{\partial{a^\ast}}{\partial{q}}$ and I don't see why.

This question is regarding a particular paper that claims a particular result that I cannot seem to follow. The paper is: Cyclic Spectroscopy of the millisecond pulsar, B1937+21 (The paper should be ...
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0answers
34 views

Partial Derivatives - Chain Rule / Polar Coordinates Problem

(please see image link below) Currently stuck on this problem involving the chain rule and partial derivatives. Having looked at the highlighted part of the solution, I can't figure out two things: ...
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1answer
25 views

Directional derivative equivalent definition

Let $f=f(x_1,\dots,x_n)$ be a scalar function defined on some open subset $U\subset\mathbb{R}^n$. Given an unit vector $v=(v_1,\dots,v_n)$ and a point $x_0\in U$, the directional derivative of $f$ at $...
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0answers
20 views

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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0answers
29 views

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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1answer
40 views

How was this integral with functions as limits solved? [duplicate]

How was this integral solved ? $$\displaystyle \frac d{dy}\int_{g(y)}^{h(y)}f(x,y)dx$$ $$=\int_{g(y)}^{h(y)}\frac \partial{\partial y} f(x,y)dx+f(h(y),y)\frac{dh(y)}{dy}-f(g(y),y)\frac{dg(y)}{dy}$$ ...
2
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1answer
26 views

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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2answers
37 views

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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1answer
30 views

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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2answers
37 views

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 ...
1
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2answers
43 views

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 ...
1
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1answer
31 views

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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2answers
46 views

Partial Differentiation of $\frac 00$

Let: $$f(x,y)=x^2y\sin\left(\frac{y}{x}\right),\ x\neq0$$ $$f(x,y)=0, \ x=0$$ Partial differentiation is obvious for $x\neq0$, however, for $x = 0$ and the derivative over $x$, one gets: $$\lim_{h\to ...
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2answers
32 views

Calculating derivative with multiple variables

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions Try to find $$\frac{\partial z^2}{\partial t^2}$$ I've tried it and only got: $$\frac{\partial z}{\...
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2answers
51 views

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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0answers
22 views

Finding theta when using polar coordinates in solving double and triple integrals

So I have a problem with finding theta. I can always see from where to where the theta is going. The problem occurs when calculating volumes. Sometimes in the solutions book, for the full circle $$4\...
2
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1answer
32 views

Prove $\iiint_V\frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}}dV$ is constant inside a ball

I have been struggling with this problem for a while: Let $V$ be the volume: $$V=\{(x,y,z)| R_1^2\leq x^2+y^2+z^2\leq R_2^2\}$$ Such that $0<R_1 <R_2$. We will define a new function $\phi(a,...
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1answer
29 views

Determining if the limit $\lim_{(x,y)\to(0,0+)} x \ln y$ exists

I am having trouble determining if the limit $$\lim_{(x,y)\to(0,0+)} x \ln y$$ exists. I tried using the path $y=kx^n$, but this gives us nothing: $$\lim_{x\to 0} x \ln kx^n=\lim_{x\to 0} x (\ln k+...
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1answer
9 views

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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0answers
33 views

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.
2
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2answers
48 views

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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2answers
20 views

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.
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1answer
21 views

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). ...
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1answer
30 views

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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1answer
35 views

Find the linearization of $f(x,y) = \sqrt{2} \cos(x) \sin(y)$ at the point $(0, \pi/4)$ [on hold]

A function $f(x,y) = \sqrt{2} \cos(x) \sin(y)$ is defined for domain $D$ including the point $P_0(0, \pi/4)$. Find the linearization $L(x,y)$ of the function and find an upper bound for the error $|E(...
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2answers
37 views

Find out whether a function of several variables is differentiable

$$ f= \begin{cases} \frac{xy}{(x^2+y^2)^\frac{1}{4}},\ (x, y) \ne (0, 0) \\ 0,\ (x, y) = (0, 0) \end{cases} $$ I need to find out whether this function is differentiable at $(0, 0)$ and $(1, 0)$ and ...
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0answers
31 views

Tangent space of matrix group

In Andrew Baker's book 'Matrix Groups', he as defined tangent space to $G$, a matrix group, at $U \in G$ as $$T_U G =\{\gamma ' (0) \in M_n(\mathbb K) : \gamma \ \text{is differentiable curve in} \ G ...
1
vote
1answer
31 views

how do you graph xyz=1

is there a graphic device that can graph xyz=1? if so, what does it look like? i tried the 'geogebra 3d calculator' and it didn't work out so well edit* thank you so much for answering! it helped me ...
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0answers
20 views

Basic Multivariable Calculus Question

My friend is studying for a calculus exam by solving previous exams. He asked me about a question which states: Show that : $$\int_{0}^{x} [\int_{0}^{v}[\int_{0}^{u}dt]du]dv = \frac{1}{2} \int_{0}^{...
1
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1answer
13 views

Finding line integral of force ${\bf F}=(x+y,y^2-x)$ along closed curve consisting of 2 piece wise regular curves.

Compute the line integral $$\oint_C{\bf F}\cdot d{\bf r}$$ for ${\bf F}=(x+y,y^2-x)$ and $C$ is the curve which begins at $(-1,0)$, proceeds along the $x$-axis to $(1,0)$ and returns to $(-1,0)$ by ...
0
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2answers
29 views

Calculate line integral L: $y=sinx$, $y=0$, $0\le x \le \pi$

There is an example to calculate the line integral $\oint_{L}P(x,y)dx+Q(x,y)dy$ The contour $L$: $y=\sin x$, $y=0$, $0\le x \le \pi$ $P(x,y)=e^{x}y$, $Q(x,y)=e^{x}$ The calculation has to be ...
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votes
1answer
32 views

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 .