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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3answers
182 views

Limit of $x^2-4xy+y^2$ as $(x, y)$ go to infinity

Can someone please tell me what the limit is of the following function as $x,y$ go to infinity? $$x^2-4xy+y^2.$$ I think it is $-\infty$, but not sure. thanks
1
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0answers
108 views

Differentiating a multivariable function

In numerical mathematics, we looked at the "Taylor series method" to construct one-step methods. Let $\mathbf{\dot{y}} = \mathbf{f}(\mathbf{y})$ be a system of differential equations. We define a ...
4
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1answer
117 views

Is this really a typo?

Let $U \subseteq \mathbb R^n$ and $F: U \to \mathbb R^m$ a function with coordinate functions $f_i$. My notes say that: If $F$ is differentiable on $U$ the Jacobian of $F$ is defined at each point in ...
3
votes
4answers
838 views

Volume of cube section above intersection with plane

Suppose we have a unit cube (side=1) and a plane with equation $x+y+z=\alpha$. I'd like to compute the volume of the region that results once the plane sections the cube (above the plane). There are ...
0
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0answers
108 views

Transforming an integral over $R^n$ to a radius and a directional vector (aka Spherical-Radial)

In several papers by John Monahan and Alan Genz there's mention of a spherical-radial transformation: $$ \int_{\mathbb{R}^n} f(\mathbf{x}) ~\mathrm{d}\mathbf{x} = \int_0^\infty ...
2
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1answer
1k views

Double integral in polar coordinates

Use polar coordinates to find the volume of the given solid inside the sphere $$x^2+y^2+z^2=16$$ and outside the cylinder $$x^2+y^2=4$$ When I try to solve the problem, I keep getting the wrong ...
0
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1answer
103 views

Stereographic projection when the “North/South Pole” is not given by $(0,…,\pm 1)$?

Straight forward enough... what if My point is arbitrary, how can I get a new stereographic projection?
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0answers
51 views

Calculate $\int_\gamma(y^2+z)\,dx + z\, dy + xy\,dz$ where $\gamma$ is the intersection between two surfaces.

The problem I have is to find the normal vextor $\textbf{N}$, for Stokes's theorem, and then determine the area of integration. Here's the question: Let $\gamma$ be the intersection between the ...
0
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1answer
37 views

Some doubts about the boundary of set $A$ in the ambient space $V$ (in the book: Multidimensional real analysis. 1)

When I reading Duistermaat J. & Kolk J's Multidimensional real analysis. 1 Differentiation.(you can read from here: Multidimensional real analysis. 1) In page 11. I can't understand the ...
3
votes
3answers
300 views

Divergence transforms as scalar under rotation in 2D + intuition

Problem is as follows: In two dimensions, show that the divergence transforms as a scalar under rotations. Aim is to determine $\bar{v}_{y}$ and $\bar{v}_{z}$, and show that ...
3
votes
2answers
103 views

Find the volume of $K=\{(x,y,z):|x-z^2|+|y-z^2|+z^2\le1\}$

How can I find the volume of $K=\{(x,y,z):|x-z^2|+|y-z^2|+z^2\le1\}$? First, $z^2\le1$ since $|x-z^2|+|y-z^2|\ge0$ This would give me $$\int \limits_{-1}^1 \left(\iint\limits_K \,dx\,dy\right)\,dz$$ ...
2
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1answer
290 views

Difference between Scalar field and a multivariable Function?

If a scalar field gives out a normal number for every orders pairs what's the difference between it and a function.
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2answers
83 views

How do we solve such an equation

I've been reading about the inverse function theorem and i tried to solve this problem that seems quite elementary: Find where $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2; f(x,y)=(2xy,x^2-y^2)$ is ...
2
votes
2answers
82 views

How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole

This seems pretty trivial but I'm not sure what to do. My coordinates are Cartesian, and I want to send point the $(x_1,...,x_{n-1},x_n)$ to point $(0,...,0,-1)$ so that all the other points are also ...
3
votes
3answers
696 views

Maximum and minimum function on circle

Find minimum and maximum value of function $f(x,y) = 3x+4y + |x-y|$ on circle $$\left\{ (x,y): x^2+y^2 = 1 \right\}$$ I used polar coordinate system. So I have $x = \cos t$ and $y=\sin t$ where $ t ...
4
votes
1answer
317 views

Leibniz integral rule confusion

Consider a vector field $\mathbf B:\mathbb R\times \mathbb R^3\to\mathbb R^3$. Let there exists some $\mathbf A:\mathbb R\times \mathbb R^3\to\mathbb R^3$ for which $\mathbf B = \nabla \times\mathbf ...
1
vote
1answer
72 views

How should I be thinking about the total derivative at a point?

I am curious as to how I should be reasoning about the total derivative of a function when evaluated at a point. I have been thinking of these objects as linear functions, which it seems to me that ...
3
votes
1answer
108 views

Dot Product/ Cross Product Proof

Let $\hat{a}$, $\hat{b}$, and $\hat{c}$ $\in \mathbb{R}^3$, using the properties of vectors, prove $$ (\hat{a} \times \hat{b}) \cdot [(\hat{b} \times \hat{c}) \times (\hat{c} \times \hat{a})] = ...
5
votes
2answers
232 views

Integral of $e^{y^2}$

Why can't be we take the antiderivative of $e^{y^2}$ with respect to y? My textbook just says it is not possible. That's it. No explanation. I could not find any one explanation on the web where it ...
1
vote
2answers
224 views

Double Integral Over Region Common to Two Circles

Evaluate $\iint\frac{(x^2+y^2)^2}{x^2y^2} dx dy$ over the region common to the circles $x^2+y^2=7x$ and $x^2+y^2=11y$.
3
votes
1answer
88 views

Proof of the potential function representation of Complex lamellar vector field

Given a continuously differentiable vector field $\bf a$, demonstrate the equivalence (iff) between the requirement that it satisfies ${\bf a}\cdot(\nabla \times {\bf a})=0$ and that it has the ...
2
votes
1answer
59 views

Change of variables in 3 dimensions

Consider the following integral: $$\int_{|x| = \epsilon} \phi(x) \frac{e^{-m|x|}}{4 \pi |x|^2} d^3x.$$ I wanna show that this integral goes to $\phi(0)$ for $\epsilon \rightarrow 0$. The idea is ...
1
vote
2answers
449 views

How can it be proved that the geometric mean function is concave?

A function $f: \mathbb R ^n \rightarrow \mathbb R $ is said to be concave if $\forall x,y \in \mathbb{R}^n, \forall \lambda \in [0,1]$ we have $ \lambda f(x) + (1-\lambda)f(y) \le f( \lambda x + ...
1
vote
1answer
47 views

Jacobian of a map into the unit circle has rank at most 1

I need to prove the following elementary fact in order to complete a proof for my bachelor thesis: Let $f:\mathbb{R}^N \to \mathbb{C}$ be a function that maps to the unit circle, i.e. ...
4
votes
2answers
2k views

Simple explanation of lagrange multipliers with multiple constraints

I'm studying support vector machines and in the process I've bumped into lagrange multipliers with multiple constraints and Karush–Kuhn–Tucker conditions. I've been trying to study the subject, but ...
1
vote
1answer
102 views

Triple integral (check solution)

The function given is $f(x,y,z) = \displaystyle\frac{1}{(x+y+z+1)^2}$ $D = \{(x,y,z) \in \mathbb{R}^3 : x \geq 0, y \geq 0, z \geq 0, x+y+z \leq 1 \}$ It seems that the domain would be the under a ...
3
votes
1answer
103 views

An inequality involving multi-index

I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this: For $x \in \mathbb{R}^{n}$ and $\alpha = ...
2
votes
1answer
398 views

Mean Value Theorem for Several Variables

Please see the mean value theorem stated here: MVT. Note that the MVT stated there requires that the segment connecting $x$ and $y$ lies in $G$. So clearly, MVT works for any pair of points if $G$ is ...
1
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2answers
44 views

Easiest way to show $ \lim \limits_{x\to 0} \frac { |x_1|^{\alpha_1} \dotsb |x_n|^{\alpha_n} } {\| x\|^p} \text { exists } \iff \sum{\alpha_i} > p$

What is the easiest way to show $$ \lim \limits_{x\to0} \frac { |x_1|^{\alpha_1} \dotsb |x_n|^{\alpha_n} } {\| x\|^p} \text { exists } \iff \sum{\alpha_i} > p, \,\,\,\,\text { for } \alpha_i ...
0
votes
2answers
282 views

Evaluate the integral by reversing the order of integration

I'm having trouble solving for the new limits when I reverse the order of integration for the integral $$\int_{0}^1\int_{x}^1{e^{x\over y}}dydx$$ If someone could help me understand how to solve for ...
6
votes
1answer
91 views

Integral in Polar Co-ordinates: Can you help evaluate it?

I have $$\int_{0}^{r_{0}}\int_{a}^{b}r_{1}e^{-\beta(r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta))}d\theta dr_{1}$$ Can anyone help me break it down for general $a$ and $b$? Alex
2
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0answers
71 views

Solution to this Poisson equation

I am struggeling with the following PDE. Does somebody here know a solution on the whole $\mathbb{R}^2$ that goes to zero for r approaching infinity? $\Delta ...
0
votes
1answer
121 views

Amazing integral with square of a series

I want to integrate the following amazing integral with Legendre Polynomials. If you need it for your solution, it might be good to know, that the series converges absolutely. I do not really have an ...
2
votes
1answer
96 views

Derive $\frac1n \|x\|_p^p \leq \|x\| \leq n^{p/2}\|x\|_p^p$ from Holder's inequality?

Given a vector $x = (x_1, \dotsc, x_n)\in \mathbb{C}^n$, I wanted to compare $|x_1|^p + \dotsb + |x_n|^p$ to $\|x\|^p$. I discovered that if $m=\max_i|x_i|$, we have $$m^p \leq \|x\|^p \leq ...
3
votes
1answer
416 views

When are $3$ vectors associative in triple cross products?

The question I am trying to show under what conditions $$\vec{A}\times(\vec{B}\times\vec{C}) = (\vec{A}\times\vec{B})\times\vec{C}.$$ NB: apologies in advance, I cannot find notation for ...
2
votes
1answer
109 views

Show that $ \lim_{(x,y) \to 0} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \alpha/\gamma + \beta/\delta > 1.$

Ted Shifrin on this site posed an interesting problem to me: show that $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + ...
0
votes
1answer
208 views

Gradient of Predictive Sparse Decomposition Cost function

I am trying to minimize the following Cost function with respect to $X_m$. $$ Energy = f(X) = \frac{1}{2}||I-\sum_{m=1}^{M}{C_m * X_m}||_2^2+\sum_{m=1}^{M}{||X_m-\phi(W_m * I)||_2^2}+\lambda|X|_1 $$ ...
5
votes
1answer
709 views

How to maximize the volume of a rectangular parallelepiped in an ellipsoid?

This question comes from an exam about 15 years ago. How to find the maximal volume of a rectangular parallelepiped inscribed in an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$? ...
2
votes
4answers
230 views

Derivative of positive part of a function

Let $f,g: A \to \mathbb{R}$ be two continuous functions defined on a compact subset $A \subset R^{2}$. Define $H:\mathbb{R}^{+} \to \mathbb{R}$ by $$H(\epsilon):=\int\int_{A}(f+\epsilon ...
0
votes
1answer
73 views

How to show this line is tangent to $f$ at point $a$?

Let $f:I\to\mathbb{R}^n$ be a differentiable function, with $f'(a)\neq 0$ for some $a$ in the interval $I\subset\mathbb{R}$. If there exists a line $L\subset\mathbb{R}^n$ and a sequence $(x_k)$ in ...
1
vote
2answers
119 views

Tetrahedral Law of Cosines, Part I

Question: Given a tetrahedral $\rho$ with faces $\Xi, \Pi, \Gamma, \Delta$ with areas $\xi , \pi, \gamma , \delta$, respectively, assign a normal vector to each face such that $\mid \mid \hat{\xi} ...
2
votes
1answer
91 views

Evaluate a triple integral

Given $f(x,y,z) = \sqrt{1+(x^2+y^2+z^2)^{\frac{3}{2}}}$ and $D=\{(x,y,z) : x^2+y^2+z^2 \leq r^2\}$, evaluate $\int\int\int_D f(x,y,z)dxdydz$. I've thought that spherical coordinates would be the best ...
5
votes
1answer
143 views

Did Brook Taylor develop his formula also in many variables by himself?

I was wondering whether Brook Taylor was also familiar with analysis in many variables at that time. I found no information about it online. Greetings Eu2718
2
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1answer
243 views

Tetrahedral Law of Cosines Proof

Given a tetrahedral $\rho$ with faces $\Xi, \Pi, \Gamma, \Delta$ with areas $\xi , \pi, \gamma , \delta$, respectively, assign a normal vector to each face such that $\mid \mid \hat{\xi} \mid \mid = ...
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votes
4answers
2k views

Double integrals over general regions

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it's easier. $$\int\int_{D}{ydA}, \text{$D$ is bounded by $y=x-2, ...
8
votes
1answer
661 views

Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
2
votes
1answer
131 views

Verify solution: Is this gradient correct?

So I want to calculate minus the gradient of $$\Phi_1=\sum_{l=0}^{\infty}f(l)r^{l}P_l(\cos(\theta))$$ where $P_n$ is the $n$-th Legendre polynomial then we have $$-\nabla ...
1
vote
0answers
38 views

Approximative solution to PDE with additional term.

I am currently struggeling with the following problem: If I have a solution to the partial differential equation $ \Delta \Psi(r,\theta) = \rho(r,\theta)$ on $\mathbb{R}^3\backslash B(0,R)$(so the ...
6
votes
1answer
281 views

Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$

Let $a$, $b$, $c$ and $d$ are non-negative numbers such that $abc+abd+acd+bcd=4.$ Prove that: $\dfrac{1}{a+3}+\dfrac{1}{b+3}+\dfrac{1}{c+3}+\dfrac{1}{d+3}\leq1$ I simplified it and it turns out that ...
4
votes
1answer
200 views

Finding the jacobian of a differential system with a piecewise function

My system: $$\frac{\mathrm{dx} }{\mathrm{d} t}=-ax^2+y^2-\gamma z$$ $$\frac{\mathrm{dy} }{\mathrm{d} t}=- h(y)-\beta y $$ $$\frac{\mathrm{dz} }{\mathrm{d} t}=x+h(y)-\beta z $$ where $h$ is the ...