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).

learn more… | top users | synonyms (1)

0
votes
2answers
185 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
85 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
votes
0answers
66 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
101 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
91 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
212 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
100 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
171 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
574 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
200 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
115 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
85 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
140 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
votes
1answer
170 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 = ...
0
votes
4answers
1k 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
611 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
123 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
273 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
173 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 ...
4
votes
0answers
243 views

Uniform Differentiability

Let $f:\mathbb{R}^n \to \mathbb{R}$ be differentiable and such that $\nabla f$ is uniformly continuous. Show that $f$ is uniformly differentiable; that is, for any $\epsilon >0$, there is a $\delta ...
5
votes
1answer
85 views

Correct Solution?

Suppose $$\omega = \left(\sum_{i=1}^{n} x_i^2 \right)^k \mid n >2$$ Find $k$ so that $$ \sum_{i=0}^{n} \frac{\partial^2 \omega}{\partial x_i^2} = 0 \, \, \, \text{for all} \, \, \, x_i$$ Proposed ...
0
votes
0answers
55 views

“Great Circle” distance [duplicate]

Given two points on a sphere, then the "great circle distance" between two points is the length of the smallest arc of a great circle containing both points. Assume that $\Sigma$ is a sphere of radius ...
3
votes
0answers
38 views

Attach term to solution of PDE(perturbation theory)

Currently I am struggeling with the following problem: Actually I have a found a solution to the PDE $\Delta \Phi(r,\theta)=f(r,\theta)$ and now I want to include a small extra term given by ...
1
vote
1answer
100 views

$\lim_{(x,y)\to (0,0)} \frac{x^m y^n}{x^2 + y^2}$ exists iff $m+ n > 2$

I would like to prove, given $m,n \in \mathbb{Z}^+$, $$\lim_{(x,y)\to (0,0)}\frac{x^ny^m}{x^2 + y^2} \iff m+n>2.$$ (My gut tells me this should hold for $m,n \in \mathbb{R}^{>0}$ as well.) The ...
2
votes
1answer
99 views

How to prove $f'(a)=0$?

Let $f:I\to\mathbb{R}^n$ be a differentiable function, where $I\subset\mathbb{R}$ is a interval. For each $c\in \mathbb{R}^n$, define $X_c=\{x\in I;\;\;f(x)=c\}$. The problem asks to show that if ...
1
vote
1answer
29 views

Question about coordinate change

Say $f$ is a function $f: \mathbb R^2 \to \mathbb R$. Can someone show me an example of such an $f$ with the property that $(\partial / \partial x)^2 f(x,y) = 0$ and $(\partial / \partial y)^2 f(x,y) ...
1
vote
0answers
851 views

Taylor's Theorem for Multivariate Functions

Please look at this theorem in Wiki regarding Taylor's theorem generalized to multivariate functions: Multivariate version of Taylor's Theorem The version stated there is one that I'm not familiar ...
0
votes
2answers
163 views

Check calculation of mean value of a vector field over a sphere

Let $E=-\nabla(\Phi)$ be a vector field, where $\Phi:\mathbb{R}^3 \rightarrow \mathbb{R}$. Is it true that the mean value $$\bar E:=\frac{-1}{V_{\text{sphere}}}\int_{V{\text{sphere}}}\nabla \Phi = ...
3
votes
1answer
82 views

Loss of direction in Gauß' theorem?

I was wondering about the following: If I have a function $\phi:\mathbb{R}^3\rightarrow \mathbb{R}$ and I want to calculate the mean value of $E=-\nabla \phi$ over a sphere, then $E$ of course if a ...
3
votes
0answers
94 views

Double Integration.

I have an integral $$\int_0 ^a\int_0 ^b\int_0 ^a\int_0 ^b \sin(x)\sin(\bar{x})\sin(y)\sin(\bar{y})f(x,\bar{x},y,\bar{y})~dx~dy~d\bar{x}~d\bar{y}$$ where $f= ...
4
votes
1answer
142 views

When are the following multiple improper integrals convergent?

This is a question from a past exam. For which $p, q\in \mathbb R$ do the following improper integrals converge? ...
11
votes
3answers
504 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
3
votes
1answer
54 views

Vector differentiation

Here is a step in a differentiation I don't understand. Let $(x^1, ...,x^k), (y^1,...,y^k)$ denote two vectors with $y^i = y^i(x^1,....,x^k)$. Given $$ y^1(0,x^2,...,x^k) = 0$$ how can I reach the ...
1
vote
2answers
191 views

Express $f'''_{xxx} and f'''_{yyy}$ in terms of $f'''_{uuu} and f'''_{vvv}$.

Let $f(x,y)\in C^3(\mathbf{R}^2)$ and let $u=x+y$ and $v=y$. Express $f'''_{xxx} and f'''_{yyy}$ in terms of $f'''_{uuu} and f'''_{vvv}$. I'm supposed to use the chain rule, how do I go about? ...
3
votes
1answer
43 views

$\nabla \varphi \overset{?}{=} \nabla \cdot \varphi \bar{\bar{I}}$ where $\varphi$ is scalar, $\bar{\bar{I}}$ is identity tensor

I am trying to determine if these two are equivalent. I have a function written with both terms, and this is the only discrepancy. The gradient increases the rank of the scalar to a vector, while the ...
7
votes
1answer
153 views

The proof of the Helmholtz decomposition theorem through Neumann boundary value problem

As you know, the Helmholtz decomposition theorem is as follows. Let $\Omega$ be open and simply connected, and bounded region in $\mathbb R^3$. Then the smooth vector field $E : \Omega \to ...
2
votes
1answer
171 views

Proper change of coordinates

It's a really easy one for you guys. I'm performing a simple cylindrical change of variable from Cartesian coordinates, but I want to write it out properly and I'm stuck with the differential ...
2
votes
2answers
222 views

A Distinction Between Different Types of Partial Derivatives

I recently noticed a subtle distinction between different types of partial derivatives: those that involve differentiation with respect to a parameter (that is, an independent coordinate) those ...
5
votes
2answers
68 views

$T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, $\operatorname{rank}(dT)=m$, show $T$ maps open sets to open sets.

Suppose $T: \mathbb{R}^n\to \mathbb{R}^m$, $n>m$, with $dT$ having rank $m$ at all points in an open set $D \subset \mathbb{R}^n$. What is a proof that $T$ maps $D$ into an open set in ...
0
votes
2answers
632 views

What is a smooth surface?

What is a smooth surface in terms of tangents and normals? I read in a book that surfaces are smooth if its surface normals depend continuously on the points of that surface. I did not understand this ...
1
vote
1answer
135 views

Intervals where a function is convex and/or concave

I find myself in need of the solution of the following problem for my work. An help is appreciated. Let $a$ be a real such that $0 \le a \le 1$. For what real values of $y$ is the function $$ f(x) ...
6
votes
1answer
321 views

Let $x = h(y, z), y = g(x, z), x = h(y, z)$ to calculate partial derivatives?

Problem: A $\mathbb{R}^{3}$ surface is defined by $F(x,\ y,\ z)=k$, where $k$ is a constant. Prove $ \frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1 $. I ...
4
votes
1answer
257 views

Derivation or Intuition of Formula for Levi-Civita Symbol

http://www.ees.nmt.edu/outside/courses/GEOP523/Docs/index-notation.pdf spouted off and threw out with no motivation $$\epsilon_{ijk} = \frac{1}{2}(i - j)(j - k)(k - i) \, \forall \, \, k \in \{1, 2, ...
1
vote
2answers
404 views

Volume bounded by the intersections of $z+x^2=4$ and $z-x^2=3y^2$

I'm having trouble with solving this question. Find the volume of the solid bounded by the intersections of $z+x^2=4$ and $z-x^2=3y^2$. Thanks in advance.
2
votes
4answers
2k views

Evaluating $\int_{0}^1\int_{0}^1 xy\sqrt{x^2+y^2}\,dy\,dx$

Calculate the iterated integral: $$\int_{0}^1\int_{0}^1 xy\sqrt{x^2+y^2}\,dy\,dx$$ I'm stumped with this problem. Should I do integration by parts with both variables or is there another way to do ...
0
votes
1answer
68 views

Proof that $\operatorname{rank}(dT)=1$ implies the image is a curve

I have a question about the proof that if the differential $dT$ of a transformation has rank 1 (2) at each point in a domain, then the image will be a curve (surface). Stated more precisely (in ...
3
votes
2answers
100 views

Double integral

Calculate the iterated integral $$\int_{1} ^4\int_{1} ^2 \left(\frac xy+\frac yx\right)\,dy\,dx$$ This is the work that I've done, but it'd lead me to the wrong answer, so either I did it completely ...
0
votes
1answer
352 views

How can I check the nature of critical point on three variable function

I have study on multivariate calculus. What is the best way to finding the nature of critical point on a real-valued three a variable function? In two variable function we can use $$D = ...