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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16 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
10 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 - ...
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1answer
27 views

Parametrization of two curves.

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 ...
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1answer
24 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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7 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 = ...
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1answer
13 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 ...
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2answers
23 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 ...
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0answers
31 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
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1answer
30 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 ...
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0answers
14 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
14 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
15 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 ...
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1answer
39 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 ...
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2answers
24 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
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1answer
11 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 ...
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0answers
16 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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33 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 ...
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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 ...
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0answers
24 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 ...
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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 ...
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1answer
24 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 ...
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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} ...
1
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1answer
19 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 ...
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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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3answers
31 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
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1answer
30 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
32 views

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

I am trying to evaluate this integral: $$\int_{0}^{1}\int_{x}^{1} y^2 \sin(2\pi \frac{x}{y})dydx$$ $$=\int_{0}^{1}\int_{0}^{1} \chi_{[x,1]}(y) y^2 \sin(2\pi \frac{x}{y})dydx$$ ...
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3answers
59 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
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1answer
67 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 ...
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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 ...
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1answer
41 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 ...
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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 $F_z(a,b,c)\neq0$ ...
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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 $$\vec{F}=\left \langle x^{2},y,z \right \rangle$$ $$D=\left \{ \left ( x,y,z \right )|x^{2}+y^{2}+1\leq z\leq 5 \right \}$$ ...
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1answer
16 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 ...
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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 ...
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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 ...
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0answers
73 views

Is there a way to evaluate this integral? What are the bounds on its solution set?

I came across this integral in a problem I was trying to solve yesterday: $$ ...
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1answer
35 views
+50

Complicated surface integral/line integral.

Problem Compute the integrals $$I=\iint_\Sigma \nabla\times\mathbf F\cdot d\,\bf\Sigma$$ And $$J=\oint_{\partial\Sigma}\mathbf F\cdot d\bf r$$ For $F=(x^2y,3x^3z,yz^3)$, and ...
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1answer
35 views

Differentiating $- \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|x - n - y|^2}{4t^2}})dt$ wrt $x$?

I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect ...
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1answer
40 views

How does Green's theorem apply here?

Let $D$ be the region delimited by $$\partial D: \begin{cases} C_1: x^2 + y^2 = 5^2\\ C_2:(x-2)^2+y^2= 1\\ C_3:(x+2)^2+y^2 = 1\\ C_4: x^2+(y-2)^2= 1\\ C_5: x^2+(y+2)^2= 1 \end{cases} $$ I've sketched ...
0
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1answer
22 views

Chain rule in partial derivatives

I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow To be more specific, it's the diferentiation of (6.9) right at the ...
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0answers
15 views

To show the inverse operator Inv is continuous

The book said we can use the identity $X^{-1}-Y^{-1}=X^{-1}•(Y-X)•Y^{-1}$ to prove the Inverse operator Inv is $C^0$. Also, assume Inv is $C^{(r-1)}$, how can I prove it is $C^r$ and $C^{(r+1)}$? ...
2
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1answer
35 views

Showing that a function is injective

I am trying to show that the following function is injective in some neighborhood of $(0, 0)$: $f:\mathbb R^2 \rightarrow \mathbb R^2$ given by $$f(x, y)=(\sin(x^3)\cosh(y), \cos(x^3)\sinh(y))$$ I ...
3
votes
1answer
19 views

Show that the vector field $\vec F=(xf(u),xg(u))$ is not conservative

I'm trying to prove that the vector field $\vec F=(xf(u),xg(u))$ with $u=xy$ is not conservative. I suppose that there is a function $\phi$ so that $\nabla \phi= \vec F$. So I need to satisfy that: ...
0
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0answers
26 views

why the concepts of partial derivatives and differentiability need a open set?

In order to define the limit of a function $f: A \subset \mathbb{R^n} \to \mathbb{R^m}$ at a point $x_0 \in \mathbb{R^n}$ we need $x_0$ to be a limit point of $A$. But in order to speak about partial ...
0
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1answer
43 views

Graph of $f(x,y) = \frac{3x^2 y}{x^2+y^2}$ near the origin

I am trying to graph the function $f : (x,y) \mapsto \frac{3x^2 y}{x^2+y^2}$ on a TI-89 Titanium. I have noticed that no matter how many times I zoom in toward the origin the graph appears identical. ...
1
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1answer
20 views

Prove Continuity of a multivarible function.

I'm trying to prove the following: Let $f:\mathbb{R^n}\times\mathbb{R} \to \mathbb{R}$ be a continuous function. We define $$F(x,t) = \int_{0}^{t}f(x,s)ds $$ Prove that F is also continuous. I ...
0
votes
1answer
20 views

Calculate the vector surface integral

Let $V=\{(x,y,z)\in \mathbb{R}:\frac{1}{4}\le x^2+y^2+z^2\le1\}$ and $\vec{F}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{(x^2+y^2+z^2)^2}$ for $(x,y,z)\in V$. Let $\hat{n}$ denote the outward unit normal ...