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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1answer
56 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
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0answers
33 views

A question about the existence of a smooth function [duplicate]

Does there exists a smooth function $f: R^2 \rightarrow R$, such that $f(x,y)\ge0$, for any $(x,y) \in R^2$, and $f$ has exactly two critical points $(x_1,y_1), (x_2, y_2) \in R^2$ with ...
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3answers
73 views

How does $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot |\vec{w}|}$ make sense?

In Multivariable Calculus, the professor said that in order to compute the angle $x$ between two vectors $v$ and $w$, we use the formula: $cos(x) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| \cdot ...
2
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1answer
44 views

Can someone verify this identity for switching the order of integration

Can someone verify that the identity $$\int_a^b \int_x^b f(x,y) \,dy\,dx =\int_a^b\int_a^y f(x,y) \,dx\, dy$$ is true? For the function $f(x,y)=x^2+y^2 $ I get the same for both integrals: $$ ...
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3answers
43 views

Circle in the plane becomes cylinder in space?

I understand how the equation for example, $x^2 + (y-1)^2 = 5$, describes a circle on a 2d plane; but I was told that in space (3d) the same equation describes a cylinder, and I don't understand that. ...
1
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1answer
88 views

Double Integral [closed]

I need help calculating these double integrals (in order to show they are not equal): $$\int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} dy\,dx \ne \int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} dx\,dy$$
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1answer
48 views

$\int_D e^\frac{x-y}{x+y}dxdy$ Where is my mistake?

I'm trying to compute $$\int_D \! \exp\left(\frac{x-y}{x+y}\right) \, \mathrm{d}x \, \mathrm{d}y,$$ where $D$ is the region $0 \leq x \leq 1$, $0 \leq y \leq 1-x$, a triangle in the first ...
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1answer
37 views

Question about the implicit function theorem

Define $f(u,v) = w$ for $u,v,w \in \mathbb{R}$, $f \in C^1$ . I'm told that $f(0,0) = 0 $ and that $af_u(0,0) + bf_v(0,0) \neq 0$. I'm asked to prove: the equation $f(x-az,y-bz) = 0 $ defines $z$ as ...
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4answers
72 views

Center of mass of $x^2+y^2 \leq z \leq h$

I'm trying to find the center of mass of this shape $x^2+y^2 \leq z \leq h$but im having difficulties founding the limits of integration. using cylindrical coordinates, $x=rcos \theta$, $y=r \sin ...
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1answer
29 views

turn $D=\{(x,y) | x \leq y \leq 2x, a\sqrt{x} \leq y \leq b \sqrt{x}, x\geq 0, 0 <a<b\}$ into a rectangle

Im trying to transform the region $D=\{(x,y) | x \leq y \leq 2x, a\sqrt{x} \leq y \leq b \sqrt{x}, x\geq 0, 0 <a<b\}$ to a rectangle by using a variable change. I'm doing this in order to ...
1
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1answer
39 views

Explain what the teacher did, convergence of improper integral

I'd like someone to explain what the teacher did, because I'm not sure I understand. Basically, the question is for which values of $p$, does the integral $$\int_{1}^{\infty} \frac{dt}{t \log ...
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2answers
29 views

Limit of multivariable function

I need to find the limit: $ \lim \limits_{P \to P_.0} \frac{x^2 y^2}{x^2 + y^2}$ where $P_0 = {0,0}$ I know that i can find the limit using polar coordinates, BUT how can i find it with $\epsilon - ...
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2answers
45 views

$f : \mathbb R^n \to \mathbb R$, what is the gradient of $f(tx)$?

Fairly simple question, suppose there is a function $f: \mathbb R^n \to \mathbb R$, and a scalar $t \in \mathbb R$. is it possible to find $D_f(tx)$ using only $t$ and $D_f(x)$? Perhaps using chain ...
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2answers
48 views

Limits in vector calc

$$\lim_{(x,y) \to (0,0)} \frac{y^4\sin(xy)}{x^2 + y^2} $$ now the textbook I'm using is very bad with giving examples - about a 150 pg text that condenses everything. So the method I first tried ...
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0answers
129 views

Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
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2answers
433 views

is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
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1answer
39 views

Integrating in spherical polar coordinates

Given a function $f(r,\theta,\phi)$ expressed in polar coordinates, would its integral over a sphere of radius $R$ centered at the origin be: $$\int_0^{2\pi}\int_0^{\pi}\int_0^R ...
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1answer
26 views

How does the transformation $u=x+y$, $v=x/y$ transform the first quadrant?

How is the region $(x,y) \in [0,\infty] \times [0,\infty]$ transformed under the change of coordinates given by $$u=x+y$$ $$v=x/y$$ Would appreciate any hints on how to find the image of such ...
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2answers
69 views

solving $ \lim_{(x,y) \rightarrow (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} $

in solving $ \lim_{(x,y) \rightarrow (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} $, my textbook says to replace $x^2+y^2$ with $r^2$, thus making the equation $ \lim_{r \rightarrow 0} \frac{\sin(r^2)}{r^2} ...
3
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2answers
63 views

Where is my mistake $\iint_{Q} (x+y)^{2013}dxdy$

I'm preparing for a calculus exam, and I tried to solve the following question. $Q$ is square $[-1,1]^2 \subset \mathbb R^2$ We are asked to evaluate $\iint_Q (x+y)^{2013}dxdy$ Here is what I did: ...
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0answers
27 views

Poisson equation on a square

Studying PDEs from the notes of my professor, and there's a part I don't understand about seeking a solution for the Poisson equation on a square. Let's start from the beginning though. We want to ...
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0answers
57 views

Is this correct and sufficient to show limit does not exist?

Find limit or show that it does not exist: $$\lim_{(x,y) \to (0,0)} \frac{ 2x^{2}y^{3/2} }{y^{2}+x^{8}}$$ using the path $x=m y^{1/4}$: $$\lim_{(my^{1/4},y) \to (0,0)} \frac{ ...
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1answer
45 views

Calculate $\int_D x^3y\ dx\,dy$

Let $D$ the bounded region by the $y$-axis and the parable $x= -4y^2 + 3$. How can I calculate the integral $$\int_D x^3y\ dx\,dy$$ I am stuck with this problem some help to solve this please.
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1answer
29 views

Optimization of parallelepiped.

Let $K \in R^3$ the ellipsoid given by the equation $ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $ with $a,b,c > 0$ , let $(x,y,z) \in K$ on the first octant, consider the ...
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4answers
47 views

Total differential of a function

Let's consider a function $$f(x,y)=\begin{cases}\dfrac{xy^3}{\sqrt{x^2+y^2}},& (x,y)\neq(0,0)\\ 0,& (x,y)=(0,0)\end{cases}$$ does it have a total differential in point $(0,0)$? I say that it ...
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1answer
45 views

Variant on divergence theorem

If I want to prove that for any scalar field $f:\;\mathbb{R}^3\to\mathbb{R}:$ $$\int_V \boldsymbol{\nabla} f\;\mathrm{d}V=\int_{\partial V} f\;\mathrm{d}\mathbf{S}$$ Can I apply the divergence theorem ...
1
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2answers
162 views

Proving inexistence of limit

Prove that limit does not exist $$\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}$$ Obviously, since it is symmetric in $x$ and $y$, classic approach of substituing $x$ as a "simple" function - linear, ...
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1answer
80 views

The chain rule: differentiation in several variables

Hi! I am trying to study for an upcoming exam by doing online problems, but this one has me completely stumped. If someone can help answer or explain how to do this problem I would really appreciate ...
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0answers
45 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
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1answer
45 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
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0answers
26 views

Prove or disprove the statement related to the definition of multivariable-differentiable function

The question: Let $f,f_1,...,f_n \; (n > 0)$ be functions from $\mathrm{D} \subset\mathbb{R}^n$ to $\mathbb{R}$ satisfying $$\left ( \sqrt{\sum_{i=1}^n x_i^2} \right ) f(\mathrm{x}) = \sum_{i=1}^n ...
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1answer
38 views

Integrate the differential form over a cardioid

$\omega=\dfrac{-ydx+(x-1)dy}{(x-1)^2+y^2}$ Calculate $\int_C\omega$ where $C...r=1+\cos\varphi$ (positively oriented) I'm still pretty lost when it comes to differential forms but as far as I ...
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2answers
76 views

Time derivative of flux

We have a time and even "position" invariant vector field and a surface. If the surface is moving with constant velocity, is the flux through the surface should constant in time? Also, is there an ...
4
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4answers
93 views

$\iint_{\mathbb R^2} \frac{dx \, dy}{1+x^{10}y^{10}}$ diverges or converges?

Question I'm trying to solve to prepare for an exam. I need to find out if $\displaystyle\iint_{\mathbb R^2} \frac{dx\,dy}{1+x^{10}y^{10}}$ diverges or converges. What I did: I switched to polar ...
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1answer
32 views

Proof of Equality with Mixed Partials

Here is the link: Hessian I understand everything but this line: $$g(x_0 + \Delta x) − g(x_0) = \frac{dg}{dx} (ξ) \Delta x$$ i.e., $$S (X_0, \Delta x, \Delta y) = \frac{∂φ}{∂x} (ξ, y_0 + \Delta y) − ...
2
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1answer
96 views

A set is compact if and only if every continouos function is bounded on the set?? [duplicate]

I was asked to prove the following statement: let $K \subseteq R^n$. show that $K$ is compact (meaning closed and bounded) if and only if every continouos function is bounded on $K$. What I did: ...
2
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1answer
51 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
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2answers
198 views

Hard Integral $\frac{1}{(1+x^2+y^2+z^2)^2}$

Prove that $\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty}\displaystyle\int_{-\infty}^{\infty} \frac{1}{(1+x^2+y^2+z^2)^2}\, dx \, dy \, dz = \pi^2$ I tried substitution, ...
0
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1answer
23 views

Direction for greatest derivative

Suppose I have a function like $f(x,y) = e^x e^y x^2 y^2$, and I want to know in which direction the derivative will grow fastest at a stationary point. $(0,0)$ is a stationary point of the example ...
0
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1answer
51 views

surface integral (curl F n ds)

Let $F$ be a vector field and let $n$ be normal vector of the closed surface $S$. Then show that $$\iint_S \mathrm{curl} \ F \cdot n\ ds=0. $$ I need help on this exercise.
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4answers
130 views

surface that is created by the intersection of paraboloid and plane

Find the surface that is created by the intersection of the paraboloid $x^2+y^2-z=0$ and the plane $z=2$. $$x^2+y^2-z=0 \Rightarrow x^2+y^2=z$$ $$z=2$$ EDIT: I had to find the area of the surface ...
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1answer
61 views

Prove that $\iint_S \text{curl }\textbf{F} \cdot d\textbf{S} = 0$ where $S$ is a sphere.

Prove without using the divergence theorem. The proof using the divergence theorem is very obvious, but I need the proof which does not rely on the divergence theorem. Thanks in advance.
2
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1answer
55 views

Spherical-Coordinate Reference Frame

my problem looks apparently easy but I can figure out a solution. I'm going through a paper about the Boltzmann equation and I got stuck with this change of coordinates. The original formula for Q ...
1
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1answer
166 views

Find the flux of the vector field

Find the flux of the vector field $F = [x^2,y^2,z^2]$ outward across the given surfaces. Each surface is oriented, unless otherwise specified, with outward-pointing normal pointing away from the ...
1
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0answers
26 views

What is the 1st derivative i.r.t. coordinates for a vector function?

For a vector function $f(x,y,z)$, we have the divergence $$\nabla \cdot f(x,y,z) = \frac{\partial{f}_{x}}{\partial x}+\frac{\partial{f}_{y}}{\partial y}+\frac{\partial{f}_{z}}{\partial z}$$ , the ...
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1answer
51 views

Find the partial derivatives of second order of $f(x,y)=\varphi(xy,\frac{x}{y})$

Ok guys, I'm given this smooth function $\varphi(u,v)$ defined in $R^2$. So that $f(x,y)=\varphi(xy,\frac{x}{y})$. I have to find all partial derivatives of second order of $f$ using the partial ...
3
votes
2answers
81 views

How to find the following limit? $\lim\limits_{t \to {\pi}/{2}}\frac{ \int_{\sin t}^{1}e^{x^2\sin t}dx}{\int_{\cos t}^{0}e^{x^2 \cos t}dx}.$

I am trying to solve the limit $$\lim\limits_{t \to \pi/2}\frac{ \int_{\sin t}^{1}e^{x^2\sin t}dx}{\int_{\cos t}^{0}e^{x^2 \cos t}dx}$$ My first method was to try with L'Hopital, i derived using ...
0
votes
1answer
30 views

Tranforming to polar co-ordinates

$$I = \int_0^1\int_0^{\sqrt{1-x^2}} xy \, dy\, dx$$ By transforming to circular polar co-ordinates, evaluate I. How do I do this? Is there a formula/strategy for doing this that works with ...
3
votes
1answer
62 views

Prove that a function is differentiable if…

I'm trying to prove that given a differentiable function $f: \mathbb{R}^2 \to \mathbb{R}^m$ in $p =(p_1, p_2) \in \mathbb{R}^2$, the function $$ g(x, y) = f(x, y) - \frac{\partial f}{\partial x}(p)(x ...
1
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2answers
39 views

“Orthonormal” parameterization of solid sphere?

The standard parameterization of the solid sphere of radius $r$ centered at the origin in $3$-space is ...