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

4
votes
2answers
94 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
0
votes
1answer
29 views

$f:\mathbb{R^N}\rightarrow\mathbb{R}$ Definition of Partial Derivative Using Limit or Epsilon

Can someone share the exact definition of partial derivative for a function $f:\mathbb{R^N}\rightarrow\mathbb{R}$ in both limit language and epsilon-delta language? In particular, I have hard time ...
3
votes
1answer
25 views

Converting Ellipse Integration Boundaries To Cylindrical Coordinates

I'm having the following integral, and I'm being asked to convert the integration boundaries to cylindrical coordinates. I've figured out that on XY-plane it's an ellipse having the following ...
1
vote
1answer
33 views

curl and stokes application

I cannot fin the flux of $$F(x,y,z)=(y^2cos(xz),x^3e^{yz},-e^{-xyz})$$ through the portion of sphere $$\Sigma = \{x^2+y^2+(z-2)^2=8, z\ge0 \}$$ I think Stokes th. must be used, so in spherical ...
0
votes
0answers
11 views

Direction of a gradient at maximizer on the boundary

Let $u \in C(\bar{B})$ where $B=B_1(0) \subset \mathbb{R}^n$ is the unit ball. Assume $u$ attains its maximum at $x_0 \in \partial{B}$ and $\nabla u(x_0) \neq 0$. What can we say about the direction ...
0
votes
2answers
26 views

Eliminate the parameter

Given the parametric equations: $x = sin(\frac{1}{2} \theta)$ $y = cos(\frac{1}{2} \theta)$ Eliminate the parameter. I am completely lost. Please help.
1
vote
1answer
54 views

Jacobian and Stokes Theorem

Let $f:U \to\mathbb{R}^{n}$ a $C^{2}$ function in the open set $U \subset \mathbb{R}^{n}$. Suppose $D \subset U$ is a compact domain with boundary $\partial D$ of $C^{2}$ class. If $f(x)=0$ for all $x\...
3
votes
3answers
65 views

Prove that $\lim_{(x,y)\to(1,1)} \frac {x}{y}=1$ by epsilon delta

How can I prove that $$\lim_{(x,y)\to(1,1)} \frac {x}{y}=1$$ By epsilon delta? I am trying and I am stuck: Proof: Suppose $\epsilon >0$ we want to construct $\delta = \delta (\epsilon ) $ such ...
0
votes
0answers
39 views

Riemann integrability question in $\mathbb{R}^2$

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...
2
votes
0answers
24 views

(Conceptual_Calculus) Differential Conditions v. Derivative Conditions

I have few questions regarding the reason we learn about the differential conditions in higher dimensions and dealing with multivariable calculus. In the context of optimization (e.g. finding ...
0
votes
1answer
19 views

Do we always have min/max with $x=y=z$ for symmetric function and constraint?

Symmetric function means that $f(x,y,z)=f(x,z,y)=f(y,x,z)=...$ For example, let $f(x)=xyz$ and $g(x)=x+y+z$ where $g(x)=1$ is the constraint and $x,y,z\geq 0$ for simplicity. Both $f$ and $g$ are ...
1
vote
1answer
17 views

Square root of a $C^2$ compact-support function is Lipschitz via eigenvalues of the hessian matrix

Let $f:\mathbb R^n\longrightarrow [0,+\infty)$ be a $C^2$ function with compact support. Prove that $\sqrt f$ is $L$-Lipschitz, with $L^2\leq \frac{1}{2}\lambda(f)$, where $$\lambda(f)=\max_{x\in \...
0
votes
0answers
14 views

gradient of total variation norm in total variation denoising

I am learning total variation denoising. The gradient of TV norm need calculated. From the link: http://www.numerical-tours.com/matlab/denoisingsimp_4_denoiseregul/ It says that the gradient is ...
1
vote
2answers
64 views

Examine where $\nabla f = (0, 0)$

Given $f: \Bbb R^2 \rightarrow \Bbb R$ defined by $$f(x) = \|x\|^4 - \|x\|^2$$ where $x = (x_1, x_2) \in \Bbb R^2$, I have to examine where $\nabla f = (0, 0)$. Approach Since $||x||$ ...
3
votes
0answers
36 views

Does symmetry of second derivatives implies continuity?

I'm trying to learn calculus of several variables, and well there's a theorem which says that if all partials up to the second order are continuous then $\frac{\partial f(x,y)}{\partial x\partial y}=\...
4
votes
1answer
140 views

Integrability question with a function on a box in $\mathbb{R}^2$ (bounty added)

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...
0
votes
0answers
37 views

How calculate intersection directly without Stokes' theorem?

Calculate the line integral directly without Stokes' theorem: \begin{gather*} \oint_\gamma \mathbf{F} \cdot d\mathbf{r} \end{gather*} \begin{gather*} \mathbf{F}(x,y,z)=(2z-3y) {\hat{\mathbf{i}}} + (3x-...
1
vote
0answers
13 views

Variant of local inversion theorem in special case

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $F(x,y)=(x+2y+x^2\ ,\ y-x^3+y^2)$. Then show that for $p_0=(4,1)$ and $p_1=(1,1)$ there exists $\delta>0$ such that for every $\vec y\in B(p_0,\...
1
vote
1answer
34 views

Calculating $\sum_{n=1}^x\frac{r^n}{n^k}$ with integrals

Through some work, I've managed to solve the following sum in the form of integrals: $$\sum_{n=1}^x\frac{r^n}{n^k}=\int_0^r\frac1{a_{k-1}}\int_0^{a_{k-1}}\frac1{a_{k-2}}\int_0^{a_{k-2}}\dots\int_0^{...
0
votes
1answer
22 views

Differentiation product of functions in multidimensional Analysis

Define $k: \mathbb{R}^d \to \mathbb{R}^{m\times m}$ such that $ k(x)=g(x)f(x)^T$, where $f: \mathbb{R}^d \to \mathbb{R}^m, g: \mathbb{R}^d \to \mathbb{R}^m$ are differentiable functions. Prove that $k$...
1
vote
1answer
43 views

Diffeomorphism onto a $k$-manifold in $\mathbb{R}^n$

If $A\subset\mathbb{R}^k$ and $B\subset\mathbb{R}^n$, with $k\leq n$, and $A$ is an open set, then for $f:A\longrightarrow B$ to be a diffeomorphism it must be bijective, continuously differentiable ...
14
votes
2answers
283 views

Daunting series of integrals: $\sum_{n=2}^\infty\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log(\frac{1-\sin x}{1+\sin x})dx$

My coleague showed me the following integral yesterday \begin{equation} I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
3
votes
1answer
53 views

Study of differentiablity of function

Study the differentiability of the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ $f(x,y)=\begin{cases} \frac{x^3+y^3}{x^2+\left|y\right|} & (x,y)\ne(0,0) \\ 0 &(x,y)=(0,0) \\ ...
1
vote
1answer
46 views

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$? Well, in the answer is no. it is written that $e^{x+y}$ for every $(x,y)$ has ...
1
vote
1answer
42 views

Complete a proof that $F(x,y)$ is contracting.

Can anyone fill in the dots in this proof? Let $D := [0,\frac{1}{2}]^2$. Show there is exactly one $(x,y)=(x^*,y^*)\in D$ such that \begin{align*} x &= \frac{x^3}{2} + y^4 + \frac{1}{4} \,, \...
1
vote
1answer
27 views

Using a gradient to calculate the minimum slope

given the function: $$z=f(x,y)=e^{-x^2-2y^2}$$ I'd like to find a point where if I were to place a ball, it would roll towards the direction $(2,1,a)$ . Also, at which point could I place the ball ...
1
vote
1answer
23 views

Local coordinates for Cylinders

Suppose point $A$ has intrinsic local coordinarcs of $(0,0)$ on a cylinder of radius $7$ and point $B$ has intrinsic local coordinarcs of $(6 \pi,4)$. Find two angles that spiral geodesics could form ...
2
votes
1answer
98 views
+50

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=\...
3
votes
2answers
56 views

Prove that a function is contractive

I'm stuck with the following. I need to prove that in $D:=[0,1]\times[0,1]$ the function $F$ is contractive, where $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined as: \begin{align} F(x,y):=(\frac{...
1
vote
1answer
21 views

Divergence of Material Derivative

Let $u : \Bbb{R}^n\times \Bbb{R} \to \Bbb{R}^n\times \Bbb{R} $ be a divergence free vector field. Then the material derivative $D $ is given by: $$ \frac { \partial u_j}{\partial t}+\sum_{i=1}^{n} ...
-1
votes
2answers
75 views

Is f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}} $with f(0,0)=0 continuous in (0,0) [duplicate]

I believe that the function: f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}}$ is continuous on the point (0,0) but i can't prove it. I know you have to choose something like $x=cy^{2}$(with c a constant) to prove ...
2
votes
2answers
46 views

Find $\lim_{(x,y)\to(0,0)} g \left(\frac{x^4 + y^4}{x^2 + y^2}\right)$ where $\lim_{z\to 0}\frac{g(z)}{z}=2.$

This limit seems different to me than all the other multi variable limits already asked on this site. Let $g \colon \mathbb R \to \mathbb R $ be such that $$ \lim_{z\to 0}\frac{g(z)}{z}=2. $$ ...
1
vote
1answer
31 views

Computing the Jacobian of the Euler equations

Given the Euler equations $$ \frac{\partial q}{\partial t}+\frac{\partial f(q)}{\partial x}=0,\qquad q=\begin{pmatrix}\rho\\\rho u\\\rho e\end{pmatrix}, \qquad f(q)=\begin{pmatrix} \rho u\\\rho u^2+...
3
votes
1answer
49 views

How is “expressing” a differential operator “in cylindrical coordinates” rigorously defined?

I'm a mathematician (with little knowledge of differential geometry) trying to study physics. One of the greatest problems is the language regarding coordinate transformations. I tend to think of such ...
1
vote
2answers
57 views

Directional Derivative help, solving for derivative = 0 when given constants

A function that is useful in studying the air flow over mountains is $$h(x,y) = \frac{h_0}{[(\frac{x}{a})^2+(\frac{y}{b})^2+1]^\frac{3}{2}} $$ where $h_0$, a, and b are all positive constants. (a) ...
1
vote
2answers
40 views

Using chain rule to find partial derivatives

Let $ r = \sqrt {x^2+y^2}$ and $\theta= tan^{-1}(y/x)$ be the usual polar/rectangular relationships. Furthermore, define $u(r(x,y),\theta(x,y)) = -sech^2(r)tanh(r)sin(\theta)$ and $v(r(x,y),\theta(x,...
2
votes
0answers
54 views

Integral involving the von Mises-Fisher distribution

I'm going quickly through the VonMises-Fisher distribution $M$ on $\mathbb S^{d-1}$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $...
0
votes
2answers
30 views

How to use the implicit function theorem in this case?

Really hit a wall with this one: Prove that the equations: $$2x+y+2z+u-v-1=0\\xy+z-u+2v-1=0\\yz+xz+u^2-v=0$$ define around $(u,v,x,y,z)=(1,1,-1,1,1)$ a single function $\phi(u,v)=(x(u,v),y(u,v),z(u,...
1
vote
1answer
49 views

Calculate an integral with limit another integral

I have a list of integrals to do with a structure similar to this one, but I don't know how to attack anyone of them. I hope you can help me doing this one to understand how to do the other ones. ...
1
vote
2answers
44 views

Finding the shap of the volume $\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{1} \left(\rho^2 \sin \phi \right) d \rho d \phi d \theta$

I need to find the shap of the volume:$$\int_0^{\pi/2}\int_0^{\pi/2}\int_0^{1} \left(\rho^2 \sin \phi \right) \,\mathrm d \rho \,\mathrm d \phi\,\mathrm d \theta$$ I thought that the shape is ...
1
vote
1answer
16 views

Gradient of function, which has codomain R^2 or bigger.

For example I have a function: $$f(x_1,x_2) = \begin{bmatrix} x_1x_2^2 + x_1^3x_2\\x_1^2x_2 + x_1 + x_2^3\\\end{bmatrix}$$ Is it possible to find a gradient of this function? Because knowing the ...
1
vote
1answer
19 views

What's the formula to map between multiindices and indices?

What is the formula to map between multiindices and indices? By multiindex, I mean a variable $I\in\mathbb{N}^d$ where $|I|=\sum\limits_{i=1}^d I_i=n$. Here, $d$ denotes the dimension. Basically, ...
0
votes
1answer
32 views

Question about calculus of variation.

What is the difference between finding maxima or mimima i.e. critical point of a function and calculus of variation?
3
votes
2answers
77 views

the uniform convergence of the sequence of functions

Let $f_1:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable function. Define the sequence of functions $f_n:[a,b] \rightarrow \mathbb{R}$ by $f_{n+1}(x)=\int_a^x f_n(t)dt,$ for each $n\ge 1$ and ...
3
votes
1answer
199 views

Theorem regarding Change of Variables in finite dimesnion

My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables ...
0
votes
1answer
39 views

Area of a domain with Stokes' Theorem

This question came up on a preliminary exam: Define $$g(s,t)=(x(s,t),y(s,t))=(\cos(s)+\cos(t),\sin(s)+\sin(t)),$$ on the region $-\pi<s<\pi$, $s<t<s+\pi$. (The function $g$ is one-...
-2
votes
1answer
69 views

Integral Optimization Problem [closed]

A butterfly is flying in a room and the temperature of this room is given by the function $$T(x,y,z) = 3x^2+y^4+2z^2$$ (in Celsius °C). The butterfly is at the point $(1,1,1)$ and she realizes that ...
0
votes
0answers
27 views

$N$-dimensional volume (of revolution)

Consider the system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ and an n-dimensional shape such that, in $\{x_{1},x_{n}\}$ (and $x_{2}=x_{3}=...=x_{n-1}=0$) it is inside the lines $x_{n}=ax_{1}+b$ and $...
0
votes
1answer
62 views

curve between two points having maximum area [closed]

Find the curve(i.e. the segment of a standard curve like circle, ellipse etc.) amongst all curves(segments) that have fixed total length, passes through $ (a,b)$ and $(c,d)$ and has maximum area ...
1
vote
2answers
51 views

Question about proof: continuity of partial derivatives implies total differentiability

I have a lack of understanding regarding this proof, and since the proof is not in English, I will simply write it down up to the point where I can't go further: Statement: Assume $U \subset \Bbb R^...