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
24 views

Want to understand this slick proof of Green's Theorem using Stoke's Theorem

I saw this short proof of Green's theorem using the general Stokes' theorem, which I suppose refers to($\oint_{\partial D}\omega=\int_D\,d\omega$). I would like to understand it better, as I find ...
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0answers
15 views

Help with Change of Variable for the function $f(x,y)=e^{[\frac{x}{2x+3y}]}$

Let $D$ be the open triangle with the sides (? I think that's what it's called in English) $(0,0), (3,0), (2,0)$. For $f(x,y)=e^{[\frac{x}{2x+3y}]}$ show that $f$ is integrable on $D$ and prove that ...
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0answers
7 views

Proving Kelvin-Stokes theorem without Green's theorem

$\iint_{\omega}\nabla\times\mathbf{F}\cdot\mathrm{d}\mathbf{\omega} =\\ \iint_{\omega}\left[\left(\dfrac{\partial F_3}{\partial y}-\dfrac{\partial F_2}{\partial ...
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0answers
8 views

Weight Modification for Computationally-Efficient Nonlinear Least Squares Optimization

There was a time where I could figure this out for myself, but my math skills are rustier than I thought, so I have to humbly beg for help. Thank you in advance. I am solving a weighted nonlinear ...
1
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3answers
42 views

If $\gamma :[a,b]\rightarrow \mathbb{R}^3$ is smooth then $\gamma(t)=x$ has finite number of solutions

Let $\gamma :[a,b]\rightarrow \mathbb{R}^3$ be a smooth curve ($\gamma$ is differentiable with $\gamma'(t)\neq \mathbf{0}$ for all $t\in[a,b]$). Show that, for $x\in\mathbb{R}^3$, the equation ...
1
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0answers
4 views

Calculating the normalizing factor in the VonMises-Fisher distribution on $S^p$

I'm going quickly through the VonMises-Fisher distribution $M$ on $S^p$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $\kappa ...
0
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1answer
16 views

For convex $f$, why is $(p,q) \mapsto q \, f(p/q)$ convex on $\mathbb{R}_+^2$?

This fact was stated in the Wikipedia article on $f$-divergences to explain why they are jointly convex.
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2answers
38 views

Are these surfaces closed?

How do I know if these two surfaces $$x^{2/3} + y^{2/3} + z^2 = 1\quad \text{and}\quad x^6 + y^6 + z^6 = 1$$ are closed without using a computer program ?
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0answers
24 views

Find the absolute max and min values of a multivariable function bounded by a circular boundary

Find the absolute minimum and maximum values of $f (x, y) = xy e^{−2x^2 −2y^2}$ on the set $\Delta = $ {$(x,y)\in\mathbb{R^2} | x^2+y^2\le1$} i know i should take the partial derivatives and set ...
0
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2answers
37 views

Does the function $d: \Re^n \times\Re^n \to\Re$ given by: $d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$ define a metric on $\Re^n?$

Does the function $d: \Re^n \times\Re^n \to\Re$ given by: $$d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$$ define a metric on $\Re^n?$ How do you go about proving this? Do I need to just ...
0
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1answer
30 views

Integral of the function $\sqrt{|y^2-x|}$ on the domain $x^2\le y\le2$, $|x|\le1$

I'm trying to solve this: Find $\iint\sqrt{|y^2-x|}dxdy$ over $D$, where $D=${$(x,y)\in\mathbb R^2| x^2\le y\le2$ and $|x|\le1$} by using absolute value definition and checking the region D I ...
0
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2answers
37 views

How do I examine f on continuity?

Let $f$ be defined as follows: $$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\begin{cases}\frac{xy^{2}}{x^{2}+y^{4}}&\text{if } (x,y)\neq (0,0)\\ 0&\text{if } (x,y)=(0,0)\end{cases}$$ How do I ...
0
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0answers
24 views

How do i prove this formula? [on hold]

How can i prove this formula? $\dot{J}(\vec{X},t)=J(\vec{X},t)\dot{}div(\vec{V},t)$, Where $J(\vec{X},t)=det(\nabla_{\vec{X}}{\lambda_{t_0,t}(\vec{X},t)})$ and $\lambda_{t_0,t}$ is the movement ...
0
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0answers
6 views

Find the maximum value of $f(x,y,z)$ on the interval $x_0<x<g^x(p)$, $y_0<y<g^y(p)$, $0<z<g^z(p)$, $p=p(x,y,z)$

First of all, sorry if I am misusing terms or any tags in the post; I am a bit out of my depths here so I'm just trying to explain things in layman's terms. Now, here's the problem: I am working on ...
0
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1answer
20 views

in which subset of $R^2$ the series is convergent?

For $(x,y) \in \Bbb R^2 $ ,consider the series $\lim_{n \to \infty } \sum_{l,k=o}^n \frac{k^2x^ky^l}{l !} $ .Then the series converges for $ (x,y)$ in 1.$(-1,1)\times (0, \infty )$ ...
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0answers
10 views

How can I prove that $\operatorname{curl}(\operatorname{curl}F)=\operatorname{grad}(\operatorname{div}F)-\operatorname{div}(\operatorname{grad}F(i))$?

How can I prove that $\operatorname{curl}(\operatorname{curl}F)=\operatorname{grad}(\operatorname{div}F)-\operatorname{div}(\operatorname{grad}F(i))$ in $n$ dimensions? I guess we need to Levi-Civita ...
2
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1answer
30 views

Question regarding curl in dimensions higher than 3

According to the wikipedia page about curl curl can be defined implicily as $$(\nabla \times \textbf{F} ) \scriptsize{\bullet} \normalsize{\hat{n}} = \lim_{A \rightarrow 0} \frac{1}{|A|} \oint_C ...
1
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1answer
28 views

A definition of the Legendre transform from Zorich

This is from exercise 8.5.5.2 from Mathematical Analysis I by Zorich. The Legendre transform of a (presumably differentiable) function $f:\mathbb R^n\to\mathbb R$ is "the transformation to the new ...
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0answers
27 views

Divergence Equation

Let $u$ be a harmonic function on the open unit disc in $\mathbb{R}^2$. Is there a vector valued function $F(x,y)$ such that $$\nabla\cdot F=|\nabla u|^2?$$ What I tried is the basic steps of ...
0
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1answer
50 views

Finding this line integral, on a sphere radius $a$

$$ c \equiv \left\lbrace\left(x,y,z\right)\quad |\quad x^{2} + y^{2} + z^{2} = a^{2}\,,\quad x + y + z = 0\right\rbrace $$ $$ \mbox{Find}\quad \int_{c}x^2 $$ This is a line integral. ...
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0answers
22 views

Harmonic function — Application of Divergence Theorem

Suppose $f$ is a harmonic function on $D=\{(x,y)\in\mathbb{R}^2: x^2+y^2<1\}$. Assume $f$ is twice continuously differentiable on $cl(D)=\{(x,y)\in\mathbb{R}^2: x^2+y^2\leq 1\}$. How do we express ...
0
votes
1answer
33 views

Show $f$ is Riemann integrable on $R$.

Let $R=[a_1,b_1]\times\cdots\times[a_n,b_n]\subset \mathbb{R}^n$. Let $f:R\to\mathbb{R}$ be a continuous function. I want to show that $f$ is Riemann integrable on $R$. I know that Riemann integrable ...
0
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0answers
12 views

Area form of $M^2 \subseteq \Bbb R^4$: does $(-1)^{i+j-1}\det_{i'j'}(n,\nu) = dx^i \wedge dx^j$?

This is a follow up question from this one. I'm asking separately since one way or another, that one is sort of answered. Now, we know that if $M^{n-1} \subseteq \Bbb R^n$, then the volume element ...
3
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3answers
39 views

Distance in 3-space between two parallel vectors

I know these two lines are parallel, but I don't know how to find the distance between them. Any suggestions? Line 1: (3, 4, 7) + {6, 2, 4}t Line 2: (-1, 5, -1) + {3, 1, 2}t I tried ...
3
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2answers
93 views

How to show that $ \int_0^x\left[\int_0^uf(t)dt \right] du = \int_0^xf(u)(x-u)du$?

I have been asked to show that $$ \int_0^x\left[\int_0^uf(t)dt \right] du = \int_0^xf(u)(x-u)du. $$ But it has not been specified whether or not $f$ is continuous or if it has an anti-derivative. I ...
0
votes
1answer
21 views

Position of vertices of right triangle inscribed on $x^2+4y^2=1$ with maximum area using Lagrange Multipliers

I am asked to find, using Lagrange multipliers, the position of the vertices of a right triangle inscribed on $x^2+4y^2=1$ that has the maximum area. The two legs of the triangle (which are not the ...
3
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0answers
41 views

Inverse Function Theorem Question.

The inverse function theorem is a general statement about finding local open sets and an inverse on those sets. The standard proof by Spivak doesn't tell you about the sizes of the sets. A different ...
0
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1answer
13 views

Rate of change for a volume

I have the following question : The radius of a right circular cone is increasing at a rate of 5 inches per second and its height is decreasing at a rate of 4 inches per second. At what rate is the ...
0
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2answers
35 views

Show $|\det(v_1,\cdots,v_n)|=\text{vol}(v_1\cdots,v_n).$

I'm trying to show that $|\det(v_1,\cdots,v_n)|=\text{vol}(v_1\cdots,v_n).$ In this case, I proved that the $|det|$ is one of the volume function of a parallelepiped (4 axioms). But I have no idea ...
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0answers
32 views

Spivak's Calculus on Manifolds - Proof of Inverse Function Theorem

I have a small confusion in a step in the proof of the Inverse Function Theorem from Spivak's Calculus on Manifolds. Theorem 2-11 (Inverse Function Theorem) Suppose that $f : \mathbb{R}^n \to ...
6
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1answer
78 views

Area form for $M^2 \subseteq \Bbb R^4$

We know that in general, given a orientable hypersurface $M^{n-1} \subseteq \Bbb R^n$, the volume form on $M$ is given by $$dM = \sum_{i=1}^n(-1)^{i-1}n_i\,dx^1 \wedge\cdots\wedge \widehat{dx^i}\wedge ...
1
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2answers
65 views

If $f$ is differentiable in $(a,b)$ then $\frac{1}{f}$ is differentiable at $(a,b)$, provided $f(a,b)\neq0$

"Suppose that $f$ is a differentiable function at $(a,b)$. Prove that $\frac{1}{f}$ is differentiable in $(a,b)$, provided $f(a,b)\neq0$" We were given the following definition of differentiability: ...
1
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0answers
30 views

Spivak's Calculus on Manifolds - Statement of Lemma 2-10 is incorrect?

In Spivak's Calculus on Manifolds, there is a Lemma 2-10 that is later used to prove the Inverse Function Theorem. Lemma 2-10 : Let $A \subset \mathbb{R}^n$ be a rectangle and let $f : A \to ...
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3answers
46 views

Whether the function $f(x,y)$ is continuous at $(0,0)$

QUESTION: $$f(x,y)=\begin{cases}x \sin \frac{1}{y} + y \sin \frac{1}{x} & \text{if } xy \not = 0 \\ 0 & \text{if } xy = 0\end{cases}$$ Show that $f(x,y)$ is continuous at ...
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0answers
23 views

How to find directional derivative of f at given p? [on hold]

Question: Compute the differential of $$(f \circ g )(a,b)=\left(\cos\left(\frac{2 ab}{\pi}\right),\sin\left(\frac{2 ab}{\pi}\right),\cos\left(\frac{2 ab}{\pi}\right)\right)$$ Could anybody tell ...
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0answers
6 views

Conditional extremes, solving $xa+yb < (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$ if.

Conditional extremes, solving $$xa+yb \leq (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$$ using lagrange multipliers.. If $\frac{1}{q}+\frac{1}{p}=1$ and $p,q>1$. This reminds me of Holders ...
-1
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1answer
31 views

How to solve simultaneous inequalities (reasked)? [duplicate]

I am doing multivariable calculus, and specifically double integrals. I am facing difficulties finding the domain of the integal, however i am given the following equations: $$1≤2x+y≤2$$ $$0≤x−2y≤1$$ ...
0
votes
2answers
32 views

Showing that the multivariate normal density integrates to 1

This is NOT the same as How to show the normal density integrates to 1?. Let $\mathbf{x} \in \mathbb{R}^d$ be a multivariate normal random vector, with $\mathbb{E}[\mathbf{x}] = \boldsymbol\mu$ and ...
0
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0answers
20 views

Intuition for second frechet derivative

I am now used to thinking of the first derivative of a map between vector spaces $f:V\to W$ in the "proper" Frechet sense, as being "the assignment to each point $v$ of $V$ of the linear map ...
1
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3answers
80 views

Hottest and coldest points on a heated circular plate (use Lagrange multipliers)

A circular plate given by the relationship $x^2 + y^2 \leq 1$ is heated according to the spatial temperature function $T(x,y) = 2x^2 + y^2-y$. Find the hottest and coldest point on the plate using ...
0
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1answer
16 views

Equiareal mapping between surfaces

I have a surface parametrized with $(u,v)$ with determinant of the first fundamenthal form $\Delta =EG-F^2=\cosh \sigma +v^2\kappa^2.$ Now I'm looking for reparametrization whose Jacobian $J$ will ...
-1
votes
1answer
42 views

Prove that $\lim \frac {\sin(y^3x)}{x^2+y^2} =0 $

How I can prove that the limit is equal to 0? $$\lim \frac {\sin(y^3x)}{x^2+y^2} =0 $$ when $x\rightarrow 0$ and $y\rightarrow 0$, Its easy to see that the limit is 0, but how I can prove it? ...
1
vote
1answer
24 views

How to find the limits or boundary of integration between two points?

Im trying to calculate the work done of a field between two points. The thing im struggling with is find the limits of integration. ie. $$\int_a^b$$ Can anyone help, if the points were A(1,2) and ...
0
votes
1answer
27 views

Find Surface Area Via a Line Integral (Stokes' Theorem)

I am trying to use Stokes' Theorem to calculate the surface area of a parametrized surface via a line integral. The surface is the part of $z= x^2+y^2$ below the plane $z=5$. I know this can be done ...
1
vote
2answers
45 views

How can I prove this limit doesn't exist?

Right now, I'm doing a question: $$\lim_{(x,y)\to(1,0)}\frac{xy-y}{(x-1)^2 +y^2}$$ I know the limit doesn't exist, but I can't figure out how to prove it. I tried putting $x=1$, and getting ...
0
votes
1answer
44 views

Calculate $\frac{\partial}{\partial x_k}(\frac 1{|x-y|^{n-2}})$ and $\frac{\partial^2}{\partial x_j \partial x_k}(\frac 1{|x-y|^{n-2}})$?

I want to take first partial derivative w.r.t. $x_i$ (for $i,j,k=1,\ldots,n$) of $$x\mapsto\frac 1{|x-y|^{n-2}},\quad x\neq y.$$ where $y\in\mathbb{R}^n$ is fixed. Can I ask here if the following ...
1
vote
1answer
24 views

Where does this come from? or how do I derive it?

$$\delta \vec{x} = \frac{\partial\vec{x}}{\partial r}\delta r + \frac{\partial\vec{x}}{\partial \theta}\delta \theta+\frac{\partial\vec{x}}{\partial \phi}\delta \phi$$
0
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0answers
19 views

How to find critical points in a cubic function in two variables?

Given a cubic function $f$ in two variables $x$ and $y$ $$ f(x,y)=\sum_{i=0}^3 \sum_{j=0}^3 k_{i,j}x^i y^j, $$ I would like to find the points ($x,y$ pairs) where $\nabla f = \mathbf{0}$. Since $f$ ...
0
votes
1answer
29 views

Converse of a theorem: If the curl of a vector field is not zero does it implies it is not conservative?

I have the following theorem: If $F$ is a vector field defined in a simply-connected open set, whose coordinate functions have continuous partial derivatives and $curl(F)=0$, then $F$ is ...
1
vote
1answer
33 views

Calculate the integral $\iint_D (y^2-x^2)^{xy} (x^2+y^2)dxdy$ on a certain region

Let $D$ be the region that's bounded by $xy=a, xy=b, y^2-x^2=1, y=x$ in the first quadrant. Calculate the integral $\iint_D(y^2-x^2)^{xy}(x^2+y^2)dxdy$. Firstly, I was able to show that the boundary ...