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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0answers
28 views

Integral of $\vec \nabla f(x)$

I was trying to prove a theorem and I came across this integral as a part of the theorem: $$ \int d^3x \left(\, \psi \vec r \,\nabla^2 \psi^* - \psi^* \vec r \,\nabla^2 \psi \right)$$ I was thinking ...
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2answers
47 views

how to show that $\{x\in \mathbb R^n: f(x)=b\}$ is closed

(1) Let $f: \mathbb R^n \to \mathbb R^m$ be a continuous mapping. Let $b\in \mathbb R^m$. Show $$\{x\in \mathbb R^n: f(x)=b\}$$ is a closed set. My thought: I want to show that the set ...
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1answer
8 views

Rulings of One Sheet Hyperboloid

Let $M$ be a hyperboloid of one sheet satisfying $x^2+y^2-z^2=1$. Show that $x(u,v)=(\frac{uv+1}{uv-1},\frac{u-v}{uv-1},\frac{u+v}{uv-1})$ gives a parametrization of $M$ where both sets of parameter ...
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1answer
13 views

Tangent plane to a surface at a point

We have a function $f(x,y) = x \sin(xy) + 2$. The equation for the tangent plane to the function $f: \mathbb{R}^2 \to \mathbb{R}$ at the point $(a,b)$ is given by $$z = f(a,b) + f_x(a,b)(x-a) + ...
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3answers
19 views

Prove that $\frac{d}{dt} (\vec{r} \cdot (\vec{r}' \times \vec{r}'') = \vec{r} \cdot (\vec{r}' \times \vec{r}''')$

I have $\frac{d}{dt} (\vec{r} \cdot (\vec{r}' \times \vec{r}'') \\ = \vec{r}' \cdot (\vec{r}' \times \vec{r}'') + \vec{r} \cdot (\vec{r}' \times \vec{r}'')'\\ =\vec{r}' \cdot (\vec{r}' \times ...
2
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0answers
29 views

Problem 10 chapter 9 from PMA Rudin

If $f$ is a real function defined in a convex open set $E\subset \mathbb{R}^n$, such that $(D_1f)(\mathbf{x})=0$ for every $\mathbf{x}\in E$, prove that $f(\mathbf{x})$ depends only on $x_2, \dots, ...
2
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1answer
27 views

Basis representation of differential-form $f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^T$

I am trying to learn differential forms. I have read some scripts about differential forms and now I am trying to solve some problems. So the problem is: given $f: \mathbb{R}^2 \to \mathbb{R}^3, ...
0
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1answer
20 views

Finding the center of mass for a centroid without a convenient symmetry axis

Find the centroid of the lamina described in polar coordinates as $\left \{ \strut \left ( x,y \right )~|~0\leq r\leq 4 \cos\left ( \theta \right ),0\leq \theta \leq \frac{\pi}{3} \right \}$ Having ...
0
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1answer
15 views

How to show map is non-singular

Let $f:\;\mathbb{R}^n\to\mathbb{R}^n$ be differentiable. Suppose that for all $x\in\mathbb{R}^n:$ $$\lVert \mathrm{D}f(x)-\mathrm{I}\rVert\leq \frac{1}{2}$$ where $\lVert\cdot\rVert$ is the ...
1
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1answer
39 views

Find the partial derivative of a sphere with equation $x^2+y^2+z^2=4$

We have a sphere with the following equation: $x^2+y^2+z^2=4$ We seek to find the partial derivative, with respect to $x$, of this equation. We think of this equation as a function of three ...
0
votes
2answers
52 views

Evaluate the limit or prove that it does not exist [on hold]

I want to evaluate $\displaystyle \lim_{(x,y)\to (0,0)}\frac{\ln(1-x^2-y^2)}{x^2+y^2}$. Any idea how to prove the answer is -1? I don't see an easy way to simplify this.
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0answers
36 views

Show $\frac{y-x}{(2-x-y)^3}$ is not integrable on $[0,1]\times[0,1]$, not invoking Fubini's theorem.

The double integral $$I = \int_{[0,1]\times[0,1]}\frac{y-x}{(2-x-y)^3} dxdy$$ does not have a finite value. The two iterated integrals have different values (Counterexample to Fubini?). Then Fubini's ...
0
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1answer
31 views

Derivative of an integral on a level set

Consider a mapping $\xi:\mathbb{R}^d\rightarrow\mathbb{R}^k$ such that $D\xi \, D\xi^T>\delta\, I_k$. Here $D\xi:\mathbb{R}^d\rightarrow \mathbb{R}^{k\times k}$ is the Jacobian. Consider a ...
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0answers
16 views

Derivative of equation in matrix form

I need to compute first derivatives of the following function $S(w)$ with respect to $w$. Then solve it. The reason behind that is to minimize $S(w)$. $S(w)=\sum_{i=1}^{n} w_i^{1/2} \bigg(y_i - ...
-1
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1answer
40 views

Parametrization of two curves. [on hold]

I have an assigment to parametrize the edge of the volume which is given by the intersection of the two curves $x^2+y^2+z^2=2$ and $z=x^2+y^2$. I really have no idea how i can parametrize this? I know ...
0
votes
1answer
45 views

Multivariable Calculus, Parametrization and extreme values

I want to find the extreme values of the function $f(x,y,z) = 2x + 2y + z$ under the constraints $x^2+y^2+z^2 \le 2$ and $x^2 + y^2 \le z$ The task is to use a parametrization of the two ...
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0answers
11 views

Transforming UV Region to XY Bounded By Hyperbolas and Lines

Suppose I have a region in the x-y plane bounded by: $y=\frac{1}{x}, y=\frac{4}{x}, y=x, y=4x$ We see that: $1\leq yx \leq 4$, and $1\leq \frac{y}{x} \leq 4$ If I let $u=yx$ and $v = ...
1
vote
1answer
17 views

Let $n>1$ and $g_1,…,g_{n-1}$ be $C^2$ scalar fields over $\mathbb R^n$ , then for any scalar field $f$ , is $\det J(f,g_1,…,g_{n-1})=0$?

Let $n>1$ and $g_i:\mathbb R^n \to \mathbb R$ be scalar field for each $1\le i\le n-1$ such that all second order partial derivatives of each $g_i$ exist and are continuous ( i.e. each $g_i$ is ...
0
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2answers
26 views

Proving that $f^2$ is differentiable given that f is differentiable at $(x_0,y_0)$

So I've tried using the definition: $f$ is differentiable at $(x_0,y_0)$ iff $$ f(x,y)-f(x_0,y_0)=\frac{\partial f}{\partial x}(x_0)\cdot x+\frac{\partial f}{\partial y}(y_0)\cdot y+o(\sqrt ...
1
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0answers
38 views

Fermat's theorem for $\mathbb{R}^n$

Suppose that $f$ is a differentiable function in an open set $E\subset \mathbb{R}^n$, and that $f$ has a local maximum at a point $\mathbf{x}\in E$. Prove that $f'(\mathbf{x})=0.$ I am sorry if this ...
2
votes
1answer
42 views

gradient of gradient is it Hessian?

Say, I have a function $f(\vec{x}) = \cfrac{1}{2}\vec{x}^{T}Q\vec{x} - \vec{b}^T\vec{x}$, where $Q$ is Symmetric Positive Definite $\in R^{nxn}$. I want to find $\nabla f(\vec{x} - \nabla ...
0
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0answers
17 views

Change of variables to polar in higher dimensions

For $x_0 \in \mathbb{R}^n$ I'm trying to apply a polar change of variables to write $$ \int_{|x - x_0| < R_0 - c_2t} f(x,t) \, dx = \int_{r=0}^{r=R_0} \int_{S^{n-1}} f(x_0 + r\omega, t)r^{n-1}\, ...
1
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0answers
16 views

Differentiation with integration region depending on $x$ to solve for decreasing energy of wave equation

I want to show that for the general wave equation $u_{tt} - \nabla \cdot (c^2\nabla u) + qu = 0, \quad u(x, 0) = \phi(x), \quad u_t(x, 0) = \phi(x)$ we have $$ E(t) = \int_{|x-x_0| < R_0 - c_2t} ...
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2answers
16 views

Using double integration to compute an average yields a different result than computing it without integration

I'm self-learning multivariable calculus and am using double integration to compute the average value of $f(x,y)$ over some region. I'm trying to solve the following simple problem using two different ...
1
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1answer
42 views

solve $\frac{\partial u^2}{\partial x\partial y}=0$

I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$. I got a particular solution, I ...
0
votes
2answers
26 views

Find $f(\Bbb R^2)$ where $f(x,y) = (e^x \cos y, e^x \sin y)$

Let $f(x,y) = (e^x \cos y, e^x \sin y)$. What is $f(\Bbb R^2)$? I know I should take $u = e^x \sin y$ and $v =e^x \cos y$ and try to find a relation but I can't find something
0
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1answer
13 views

Taking derivative of energy of wave equation

Consider the variable coefficient, real valued wave equation $$ u_{tt} - \nabla \cdot (c^2 \nabla u) + qu = 0, \quad u(x,0) = \phi(x), \quad u_t(x, 0) = \phi(x), $$ where $c, q \geq 0$ depend only on ...
0
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0answers
18 views

Negative eigenvalue of a hessian matrix entails a local decrease in function value?

I was reading up on non-convex optimization, and I can across this sentence: "Since Hessian(f(w)) has a negative eigenvalue, there is always a point that is near w which has smaller function value" ...
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0answers
36 views

How to prove the following questions by IBP? (Integrated By Parts) [on hold]

So this is the question that I have to solve. I know this is related to IBP, but Have no idea how to start and prove... need help
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1answer
29 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
-2
votes
1answer
29 views

Infimum and supremum of two variable function [on hold]

How can I find the infimum and supremum in $\mathbb{R}^{2} $ of this function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$? Thanks EDIT: Forgive me if I did not add my thoughts but I did not know where to ...
2
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0answers
25 views

singular $1$ cube - Boundary of $2$ chain

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
0
votes
1answer
28 views

Finding correct variation for $\rho$ in spherical coordinate integration

I am having some trouble and looking for help on calculating the moment of inertia about the z axis of the region bound by the cone $z=\sqrt{3(x^2+y^2)}$ and the sphere $x^2+y^2+z^2=a^2$ if the ...
0
votes
1answer
25 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
0
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2answers
45 views

Method of characteristics - finding the particular solution using initial conditions

I am trying to use the method from my previous question to solve this PDE: $$ 3u_x + 2u_t = \cos x $$ with initial condition $u(x,0) = x^2$. So I need to solve these: \begin{align} \frac{dx}{ds} ...
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1answer
20 views

Computing Gauss's of a sphere

The vector field given as $\vec{F}=\frac{\left \langle x,y,z \right \rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$ The region $D=\left \{ a^{2}\leq x^{2}+y^{2}+z^{2}\leq b^{2} \right \}$ I've some ...
0
votes
1answer
24 views

How to find centroid of this region bounded by surfaces

I am having difficulty find the centroid of the region that is bound by the surfaces $x^2+y^2+z^2-2az=0$ and $3x^2+3y^2-z^2=0$ (lying above $xy$ plane, consider the inner region). I know the first ...
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votes
3answers
33 views

Intersection of 2 planes - find vector of intersection

I was trying to figure out the curve of intersection of these 2 planes: $$3x - y + z = 4 $$ $$ y + z = 2.$$ I realize it will be a straight and not curved, and feel like I should be able to do the ...
1
vote
1answer
31 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
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1answer
33 views

Evaluate $\int_{0}^{1}\int_{x}^{1} y^2 \sin(2\pi \frac{x}{y})dydx$

I am trying to evaluate this integral: $$\int_{0}^{1}\int_{x}^{1} y^2 \sin(2\pi \frac{x}{y})dydx$$ $$=\int_{0}^{1}\int_{0}^{1} \chi_{[x,1]}(y) y^2 \sin(2\pi \frac{x}{y})dydx$$ ...
2
votes
3answers
60 views

Why doesn't the limit $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ exist?

Why is this limit non-existant? $\lim_{(x,y) \rightarrow (0,0)} \frac{ e^{x+y} - x - y}{\sqrt{x^2 + y^2}}$ I can't seem to find $2$ different paths that would show it is non-existant.
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0answers
25 views

Change of variables for path integral.

Let $G=C^\infty([0,1];\mathbb{R}^d)$ be smooth paths, then for the path $A\in G$, consider the translation operator from $G$ to itself $T_A:G\to G$ $$T_A(g)(t):=g(t)+A(t).$$ Does there exist a ...
4
votes
1answer
70 views
+250

Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range

I was studying the Inverse function theorem when I came across the following problems : (Let the closed set $V$ i.e the range have non-empty interior) Does there exist a continuous onto ...
1
vote
1answer
13 views

When can a function have its variables seperated

Suppose I have a function $f(x,y,z)$. I need to know when one can write it as $$f(x,y,z)=a(x)\cdot b(y) \cdot c(z)$$ where $a, b, c$ are functions. I don't want to know what they are, but just whether ...
2
votes
1answer
42 views

Definition of partial derivatives from Rudin's PMA

It's the definition of partial derivative from Rudin's PMA. Why he consider $(25)$ for real functions $f_i$? What about if $f_i$ in $(25)$ replaced by vector-valued function ...
1
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0answers
14 views

Generalize Implicit Differentiation to find Tangent Plane

For a function $F(x,y,z)$ with $(a,b,c)$ on the level surface $F(x,y,z)=k$, where $F(x,y,z)=k$ defines $z$ implicitly as a function of $x$ and $y$. Using the chain rule, assuming $F_z(a,b,c)\neq0$ ...
0
votes
0answers
34 views

evaluate this region using gauss's theorem (only using the triple integral 'part')

Evaluate $$\iiint _{D}\vec{\nabla} \cdot\vec{F}\,dV$$ with $$\vec{F}=\left \langle x^{2},y,z \right \rangle$$ $$D=\left \{ \left ( x,y,z \right )|x^{2}+y^{2}+1\leq z\leq 5 \right \}$$ ...
0
votes
1answer
19 views

Vector Valued Functions: Parametrize the intersection of 2 surfaces w/ trigonometric functions

The question asks: Parametrize the intersection of the surfaces using trigonometric functions. $$y^2-z^2=x-6$$ $$y^2+z^2=81$$ $\mathbf{r}(t)=$ ____ My first step was recognizing ...
0
votes
0answers
31 views

The derivative of a function of multiple variables

I am trying to understand a step in the theory section of my differential equations textbook. The author writes, For example, suppose we transform the first order differential equation ...
0
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0answers
17 views

Can an iterated integral over a box R ={(x,y,z)|x∈[0,a], y∈[0,b], z∈[0,c]} be expressed in eight different ways?

this is my first time on stack exchange so sorry if I am not following any guidelines. I received this exact question on a midterm and answered yes, it is possible, which was considered wrong on the ...