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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17 views

How can I see that the image of an open subset of an open set under a function is an open set?

Let A an open set of ℝⁿ and ƒ:A→ℝᵐ of class C'(A). If for some P₀∈A, ƒ'(P₀) is surjective, prove that exists δ>0 such that B$_{δ}$(P₀)⊂A and for all open subset Ω⊂B$_{δ}$(P₀) it holds that ƒ(Ω) is ...
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2answers
33 views

Order of integration in spherical coordinates

I do not seem to understand the logic behind choosing the correct integration bounds and integration order when dealing with spherical coordinates. For example, consider the following problem: Let $...
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2answers
112 views

For $\phi(x,y)=(x,y+\psi (x))$ with $\psi:\Bbb{R}\to \Bbb{R}$ integrable, show that $\phi (B)$ is measurable for every box $B\subset \Bbb{R}^2$

Let $\psi:\Bbb{R}\to \Bbb{R}$ be integrable and define $\phi:\Bbb{R}^2\to \Bbb{R}^2$ by $\phi(x,y)=(x,y+\psi (x))$. Prove that for every box $B\subset \Bbb{R}^2$, $\phi(B)$ is measureable and $v(\phi (...
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0answers
30 views

How to bound difference of convex optimization problems, when the closed-form solution doesn't exit?

Let $z = \langle z_1 ,...,z_m \rangle$ where $z_i \in \mathbb{R}^d$, we define $g(z):= \arg \min_{y \in \mathbb{R}^{d}} \sum_{i=1}^{m} \parallel z_i -y \parallel_2$. We say $z \Delta z'=1$, if $z'$ ...
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0answers
25 views

The ability to solve the multivariate nonlinear equations

For m nonlinear polynomial equations with n variables and the highest degree 3, how is the current ability to solve such equations? In the webpage of IBM cplex, it says that: IBM ILOG CPLEX ...
2
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1answer
32 views

Why the norm in the definition of differentiability?

A function $f: \Bbb R^m \to \Bbb R^n$ is differentiable at $x_0$ iff $$\lim_{h \to 0} \frac{\|f(x_0+h)-f(x_0)-J(h)\|_{\Bbb R^n}}{\|h\|_{\Bbb R^m}}=0$$ Is there any particular reason we use the norm ...
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1answer
82 views

Proof of equivalence of two ways of calculating directional derivative

I am seeking the connection between two formulas that I saw to compute the directional derivative of function $f$ in the direction of a vector $\vec v$. One of them is : $$\nabla_{\vec v} f(\vec x_0)...
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0answers
27 views

Limits in polar coordinates

I cannot understand why the following statement is false: "Let $f$ be a function defined as $f:R^2\to R$ such that $f(0,0)=1$. If for all $\varphi\in[0,2\pi[$ fixed we have $\lim_{r\to 0}f(rcos(\...
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2answers
40 views

Derivative in spherical coordinates

Can someone please explain where this comes from? My textbook only says that is a derivative in spherical coordinates. (r is a position vector and $U$ is the potential energy). $-\dfrac{\partial U}{\...
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0answers
36 views

Integral equality $\int_{\partial\mathbb R_+^n\cap B^n(0,1)}\frac{|y|}{(|h|^2+|y|^2)^{\frac n 2}}dy$

Why is $\displaystyle{\int_{\partial\mathbb R_+^n\,\cap\, B^n(0,1)}\frac{|y|}{\left(|h|^2+|y|^2\right)^{\frac n 2}}dy=C\int_0^1\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{\partial B^{n-1}(...
3
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0answers
131 views

How to handle this integral? [closed]

How to deal with this integral ? $$ \int_{a}^{b}\cdots\int_{a}^{b} \frac{\mathrm{d}x_{1}\,\mathrm{d}x_{2}\cdots\mathrm{d}x_{n}} {1 + x_{1}^{2} + x_{1}^{2}x_{2}^{2} + x_{1}^{2}x_{2}^{2}x_{3}^{2} + ...
0
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0answers
42 views

How would i go about solving these such problem?

The problem is for Vector Calculus. I am not sure what this question is asking. $\text{a) Assume }f\text{ is of class }C^2.\text{ Show that }\nabla\times\nabla f = \vec{0}.\\\text{b) Is }\mathbf{F}...
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0answers
49 views

Partial derivative of vector intercepting a plane

I was reading a paper that describes the partial derivatives of a range $\rho$ that intercepts an arbitrary surface, where $\rho = |\bar{r}_t - \bar{r}_{bf}|$. The author described the influence of an ...
3
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1answer
34 views

Describe the motion of the path

Describe the motion of the path $$ \mathbf{r}=3\cos{t}\mathbf{\hat{i}}+4\cos{t}\mathbf{\hat{j}}+5\sin{t}{\mathbf{\hat{k}}} $$ The answers is: Path: the circle of intersection of the sphere $x^2+y^2+...
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0answers
35 views

How can I know when I can and can't integrate from $0$ to $2\pi$?

I'm taking vector calc right now (maybe the same as Calc 3?) and I think I'm forgetting something I used to understand. I know sometimes when you are integrating in polar coordinates (and probably ...
0
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1answer
22 views

Find an equation of the tangent plane to the given surface

Question: Find the equation of the tangent plane to the surface with equation $z = 3y^2-2x^2+x$ at the point $(2,-1,-3)$. My attempts: $\nabla f_x$ $= -4x+1$ $\nabla f_y$ $= 6y$ Setting up the ...
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0answers
15 views

Bounded parts of a function in limit study

I'm studying limits in functions with 2 independent variables. I have this solved limit: $$\lim\limits_{(x,y) \to (0,0)} \frac{-3x^2y}{x^2+y^2} = \lim\limits_{(x,y) \to (0,0)} -3y\frac{x^2}{x^2+y^2} =...
1
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1answer
47 views

Calculating the divergence of the Gravitational field $\nabla \cdot \vec{F}$

I want to calculate the divergence of the Gravitational field: $$\nabla\cdot \vec{F}=\nabla\cdot\left( -\frac{GMm}{\lvert \vec{r} \rvert^2} \hat{e}_r\right )$$ in spherical coordinates. I know that ...
3
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1answer
53 views

Linear Regression with Multiple Targets Derivation

I'm working through the derivation for the weights solution in multivariate linear regression, but keep getting tripped up when I try to solve for the case of multiple targets. Starting with the ...
0
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1answer
13 views

Taylor Series function seperation.

Say we have a function $$ F= \frac{g}{h} $$ And we want to expand it with Taylor series keeping only second grade terms. How do we know when to expand $F$ directly or $g$ and $ \frac{1}{h} $ ...
0
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0answers
16 views

Prove that $\lim_{x_2 \rightarrow a_2} \lim_{x_1 \rightarrow a_1} f(x_1, x_2) = \lim_{x_1 \rightarrow a_1} \lim_{x_2 \rightarrow a_2} f(x_1, x_2) = c$

Prove that if $a \in \mathbb{R}^2$ $D \subset \mathbb{R}^2$ $\exists{r>0}\left(B(a,r) \setminus \{a\} \subset D\right)$, where $B(a,r)$ is a ball of radius $r$ centered in $a$ $f : D \rightarrow ...
1
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1answer
48 views

$f$ satisfies Laplace equation $f_{xx}+f_{yy}=0$ but is not twice continuously differentiable

I know that harmonic function is defined as a real-valued function of $x$ and $y$ such that 1) it is twice continuously differentiable, 2) it satisfies Laplace equation in its domain, $f_{xx}+f_{yy}=0$...
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0answers
12 views

How to sketch functions in polar and spherical coordinates by hand on paper?

I've been practicing drawing surfaces in different coordinates. I can do the easier ones but no I am completely stuck on the following two: Say we define spherical coordinates as follows: so that $...
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0answers
31 views

seemingly simple stokes theorem problem please help

Find $\int\vec{F}\cdot \mathrm{d}\vec{r}$, where $C$ is a circle of radius $1$ in the plane $x+y+z=7$, centered at $(3,3,1)$ and oriented clockwise when viewed from the origin, if $\vec{F} = 4y\vec{i} ...
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1answer
27 views

help with stokes theorem problem

Suppose $F=(5x−5y)\hat{i}+(x+2y)\hat{j}$. Using Stokes' Theorem Find the circulation of $F$ around the circle $C$ of radius $7$ centered at the origin in the $yz$-plane, oriented clockwise as viewed ...
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1answer
22 views

Find function f(a,b) where f(0,b) = 0, f(1,b) = 1 and f(a,b) = 0.5

I am in search of a formula for a problem I encountered. Simply stated, I am searching for a continuous formula where the following conditions apply: I search for a continuous function $f(a,b)$ in ...
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0answers
24 views

Lagrangian Multipliers exercise

Let $M = \{(x, y, z) \in {\rm I\!R}^3 : F(x,y,z) = 0\}$ and let $F(x,y,z) = (3x^2z + y^2 + z^3-1, \, x + z-1)$ . Does the function $f(x, y, z) = x$ have any extrema in $M$? We are asked in advance ...
2
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0answers
70 views

Finding the $\lim\limits_{(x,y)\to(0,0)}\frac{x^2y+y^2x}{x^3-xy+y^3}$

I'm trying to find the limit (if it exists) of the following function of two variables: $$\lim_{(x,y)\to(0,0)}\frac{x^2y+y^2x}{x^3-xy+y^3}$$ So far I tried to prove that it doesn't exist by finding ...
1
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1answer
38 views

(Closed) Line integral of Conservative Field.

Suppose we have a conservative Field $ \vec F: D' \subseteq R^2 \rightarrow R^2$ where D is a set of points inside a closed curve (for example all the points inside a circle). Say we have subset of D',...
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1answer
32 views

Show that $z = \ln (x^2+y^2) +2\tan^{-1}(y/x)$ satisfies the laplaces's equation.

I can get to the first partial derivative of $\partial z / \partial x$ of course, but after that I get this, which I don't know how to differentiate further... I am talking about c part of the ...
2
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1answer
18 views

Jacobian determinant of a map?

For $m,n\in \mathbb N$, let $f$ is the map given by $$\begin{align} f: & \quad \mathbb R^m \times \mathbb R^n \longrightarrow \mathbb R^m \times \mathbb R^n \\ & (x,y)\mapsto f(x,y) = (x+x',...
2
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0answers
31 views

Jacobian matrix of the parametrization of (part of) a ball

I read (in E. Sernesi, Geometria 2) that the function $\varphi:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ defined by $$\varphi(\theta_1,\ldots,\theta_n,r)=\left(...
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0answers
34 views

Divergence Theorem with singularity

Let $x^2 + y^2 + z^2 = 1$ be a sphere. If there is a vector field with a component $z/(x^2 + y^2)$, for example $F(x,y,z) = (...,..., z/(x^2 + y^2))$, how could I "take" (isolate) this singularity ...
0
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1answer
23 views

approaching a limit in 3-d from different lines

Given $\lim_{x,y\to 0,0}$ $\frac{xy^4}{x^2+y^8}$ The answer is 0. I have no problem obtaining zero when approaching along the x and y axis however: approaching along the line x= y^4 I get: $\lim_{Y^4,...
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0answers
38 views

Uniform Continuity

This question has three parts. a) Difference between continuity and uniform continuity b) Geometrical meaning of uniform continuity c) Correct the example Definition of Continuity of a function ...
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0answers
15 views

Obtaining a bound for the Jacobian in a change of variables.

Suppose I have a diffeomorphism which I will use to make a change of variables for an integral for which I have to obtain an upper bound. The definition of the diffeomorphism $H$ is defined as follows ...
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0answers
16 views

Jordan measurability and areas of subsets

I want to know what is meant that a subset $S\subset \mathbb{R^2}$ is (i) Jordan measurable? (ii) of area zero? I think that I have solved (i): Our subset in the cartesian plane must be a bounded ...
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1answer
19 views

Question regarding the proof of the directional derivative

Proof of the directional derivative I have a question regarding this proof. I understand the part where they use the chain rule, but how can the point (x,y) be included? The result should just be ...
0
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1answer
21 views

Double integral over type II region to type I region

I have to evaluate the double integral: $\displaystyle I=\iint_D dxdy$, where $D=\{(x,y)\in\mathbb{R}^2 : -2 \leq y\leq 1 , y^2+4y \leq x \leq 3y+2 \}$ (region of type II). I've found that $\...
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2answers
31 views

Higher derivatives of inverse functions (Multivariable Calculus)

Given the function $$ (u,v) = f(x,y) = (x + y, x^2 - y^2) $$ I would like to compute the second partial derivative of $x$ with respect to $v$, at the point $(u,v) = (2,0)$. To calculate the ...
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0answers
38 views

Prove or disprove point is a limit point [closed]

I'm having a bit of trouble with this, so any help is appreciated. Given the function $$f(x,y)=\frac{\sin(2x)-2x+y}{x^3+y},$$ is $(0,0)$ is a limit point of the function? I need to prove whether ...
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2answers
48 views

Solving $\int \int_{D} x dxdy $ using coordinate change, where $D$ is a region of the plane.

$\int \int_{D} x dxdy $, where $D$ is a region of the plane. I know how to calculate directly using cartesian coordinates, but in this exercise I have to do a change of variables. Consider $D$ the ...
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0answers
11 views

Second order chain rule with bi-variate function

I am building an utility model, and I am confused how to make assumptions about second order equations. The First order condition is simple, it is the problem of $$\max_{M} u(H(M),M)$$ where $H(M)$ ...
1
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1answer
33 views

Taylor expansions to approximate multivariable functions

The way this question is phrased is confusing me more than the question itself, so I will quote it how it is written in my book: "Using Taylor's theorem, find linear and quadratic approximations to ...
1
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0answers
51 views

Harmonic Function - Multivariable calculus

One more exercise I stepped at while strolling through papers and journals for my preparation on the semester exams for multivariable calculus. Let $D=\{(x,y): x^2 + y^2 \leq 1\}$ A function $f:D \to ...
6
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3answers
86 views

Solving $\int_0^1 \int_0^x x \sqrt{x^2+3y^2} \,dy\, dx $

Solving $$\int_{0}^{1} \int_{0}^{x} x \sqrt{x^2+3y^2} \,dy\, dx $$ I tried doing this change of variable: $(x,y) = (u, \frac{v}{\sqrt{3}}) $ So the Jacobian is: $\frac{\sqrt{3}}{3}$, and the integral ...
2
votes
1answer
22 views

Partial derivative of an inner product and a linear transformation

Let $u(x)$ be a self adjoint operator on $R^n$, and $\langle\underline{\ },\underline{\ }\rangle$ the usual dot product. I have to show $f(x)=\langle x,u(x)\rangle$ is differentiable over all of $R^...
2
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1answer
45 views

Integral of delta function and the constant for fund. solution to laplace's eq

When finding the fundamental solution to Laplace's eqn, i.e. $G$ such that $\Delta G = \delta$ a constant has so be solved for. How I have seen this done is by finding $c$ so that $\int_{D(0,\epsilon)...
1
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1answer
33 views

Implicit function theorem, beginner question

I'm revising things for the end-term semester exams, and Implicit Function Theorem is something that we studied for 1 month (theoretically), with every possible alternation on the theorem and it's ...
6
votes
5answers
167 views

Is $f(x,y) = f(\mathbf{x})$ abuse of notation?

A scalar function $f(x,y)$ is often written as $f(\mathbf{x})$, where $\mathbf{x} = (x,y)$, but as far as I know, there is a difference between the scalar function inputs $(x,y)$ and the vector input $...