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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7 views

Using Green's Theorem to Express the Integral $I=\int_C (Pdx+Qdy)$ as an expression of $I_i=\int _{C_i} (Pdx+Qdy)$

Let $p_1,...p_n$ be points in $\mathbb{R}^n$. Let $P(x,y), Q(x,y)$ be functions with continuous derivatives in $ D=\mathbb{R}^2\setminus\{p_1,...p_n\}$ such that $Q_x-P_y=1$ for all $(x,y)\in D$. For ...
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2answers
19 views

Divergence change of variables (to polar)

I would wish to simplify this integral $$\oint_{\partial D}(f\nabla f)\cdot\hat{n}\,ds$$ in terms of a line integral of $g$ on $[0,2\pi]$ where $g(\theta)=f(e^{i\theta})$. Background info: ...
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1answer
23 views

Find the absolute max and min of a multivariable function on a bounded by a circle?

So i do understand everything up the square rectangle, in the photo here i mean, how did he come up with $(±2,0), (0,±1)$ is it because of $g(2cos x, sin x)$ and if that is the case why would he ...
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0answers
13 views

Integral of a $C^{\infty}$ vector field is independent from the parameter

Given a vector field $F \in C^{\infty}$, let $\Sigma_a:=\{ (x,y,z) \in \mathbb{R}^3 | z=a(1-(x^2+y^2)), x^2+y^2 \leq 1 \}$ be a surface and let $n=(n_i)_{i=1,2,3}$ be the normal unit vector s.t. ...
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0answers
38 views

Intersection of a smooth plane curve and a circle

Let $\gamma(t)=(x(t),y(t)):[0,2\pi] \rightarrow \mathbb{C}$ be a simple and closed $C^1$-curve. Prove that there is a small circle that intersects $\gamma$ only at two points?
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2answers
44 views

Orientability of surfaces

How to prove that a surface is orientable? Is it true that the union of two orientable surfaces is orientable? How to prove that? For example, is the union of the hemisphere $$z = \sqrt{1 - x^2 - ...
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24 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 vertices $(0,0), (3,0), (0,2)$. For $f(x,y)=e^{[\frac{x}{2x+3y}]}$ show that $f$ is integrable on $D$ and prove that $\iint_Df(x,y)dxdy=6\sqrt{e}-6$. I was able ...
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2answers
25 views

Parametric equations for intersection between plane and circle

So I was looking at this question Determine Circle of Intersection of Plane and Sphere but I need to know how to find a parametric equation for intersections such as these. My particular question is ...
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0answers
30 views

Work done in a conservative vector field

If my vector field is: $F=(1-\frac{x}{x^2+y^2})i-(\frac{y}{x^2+y^2})j$ How would I go about finding the work done between A(3,2) and B(4, -3)? I have proved that the vector field is conservative: ...
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1answer
144 views

Intuitively what is the second directional derivative?

I'm thinking that the second directional derivative, if both dd's are evaluated in the same direction, will just give you the concavity (the second scalar derivative) in that direction. Is that ...
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2answers
925 views

Multivariable/Vector Calculus Textbook Recommendation Please!

S.E friends, I am a college sophomore with a major in mathematics. I am trying to self-study multivariable and vector calculus (they means the same, right?) and prepare for Summer course on ...
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1answer
29 views

How to prove coercivity

I have a problem in understanding how to prove if a function is positive or negative coercive. I understood the definition of coercivity, which is: $$\lim_{||x|| \to +\infty}f(x) = +\infty$$ However, ...
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1answer
61 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 ...
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2answers
63 views

Does the function $d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$ define a metric on $\mathbb{R}^n?$

Does the function $d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by: $$d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$$ define a metric on $\mathbb{R}^n?$ How do you go about ...
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0answers
7 views

Calculating base radius of circular cone with 2 unknown dimensions

Okay, so I need to know the base radius of a circular cone with a Slantheight of 9cm, no other information is given.
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2answers
28 views

Leibniz Rule for Integrals

I am studying for a calculus exam and I came across this question I can't get the correct answer. Use Leibniz's rule for integrals to determine the derivative of the function $I$ defined by ...
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1answer
36 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 ...
2
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1answer
455 views

Conservation of momentum for nonlinear Schrödinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
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0answers
8 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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1answer
1k views

Integrating a solid using cartesian, cylindrical and spherical coordinates

The region $W$ is the cone shown below (see image). The angle at the vertex is $π/3$, and the top is flat and at a height of $7\sqrt{3}$. Write the limits of integration for $\int_W dV$ ...
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3answers
48 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 ...
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0answers
12 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 ...
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2answers
137 views

How to transform between two layout forms of matrix calculus?

I'm trying to derive a very simple matrix derivative: Take the derivative of $\operatorname{Tr}(A' X)$ with respect to $X$. However, I got two different answers by following different methods. ...
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2answers
1k views

Finding dimensions of a rectangular box

Find the dimensions of a rectangular box without a top, of the maximum capacity with a surface area of $108 \, cm^2$. This is my attempt at solving the problem : If $x,y,z$ are the dimensions of ...
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1answer
23 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
5 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 ...
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0answers
49 views

Global extremum when the constraint is not compact?

When the constraint is compact, the function must have both a global maximum and a global minimum somewhere in the constraint. However, if the constraint is not compact, the global extremum may not ...
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1answer
19 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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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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2answers
39 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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27 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 ...
3
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2answers
4k views

Finding the absolute max and min of a function bounded by a domain D.

The task is to find the absolute max and absolute min of this function: $f(x,y) = 4xy^2-x^2y^2-xy^3$ on the domain D: $D={(x,y) | x\ge0, y\ge0, x + y \le 6}$ So I get the partial derivatives of ...
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1answer
32 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 ...
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1answer
30 views

Scalar property of $ C(\Omega)=\sum_{|\alpha|\leq m}\color{blue}{\big|\Omega\big|^{\dfrac{2|\alpha|-n}{n}}} \int_{\Omega}|D^\alpha f|^2\ dx $

This is closely related to a previous question: Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$ This question focuses on the direct calculation (by change of ...
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0answers
29 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 ...
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2answers
43 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 ...
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3answers
81 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 ...
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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 ...
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0answers
7 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 ...
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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 ...
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1answer
30 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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2answers
1k views

Maximize the volume of a rectangular box in the first octant with one vertex in the plane $x+2y+3z=3$

Find the volume of the largest rectangular box in the first octant with the three faces in the coordinate planes and one vertex in the plane $x+2y+3z=3$. Now I know that $V=xyz$ and I have set ...
2
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1answer
31 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 ...
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1answer
52 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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4answers
97 views

Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
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3answers
89 views

Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
0
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1answer
31 views

What is the absolute maximum and minimum values of $f(x,y)=x^2-2x+2y$?

Find the absolute maximum and minimum values of $$f(x,y)=x^2-2x+2y$$ on $$d=\{(x,y)\in\mathbb{R}^2\ |\ 0 \le x \le 3, 0 \le y \le 2\}$$
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1answer
56 views

Find the absolute maximum and minimum values of $f(x,y)=(x-y)(1-x^2-y^2)$ for $x^2+y^2\le1$

Find the absolute maximum and minimum values of f on the set D: $f(x,y)=(x-y)(1-x^2-y^2)$ $D=\left\{(x,y) \mid x^2+y^2\le1\right \}$ Can someone help me resolving the system of partial ...
1
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1answer
25 views

Find the absolute maximum and minimum for $F(x,y)=x^2+3y^2-y$ with the condition $x^2+2y^2 \leq 1$

Find the maximum and minimum of the function : $$F(x,y)= x^2 +3y^2-y$$ with the condition : $$x^2+2y^2 \leq 1$$ Can this question be solved by calculus or by some other way?
3
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2answers
94 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 ...