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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The rank of a block matrix

Say I have a block matrix of the form $$\left( \begin{matrix} A & 0 \\ B & C \end{matrix}\right)$$ Is there a formula for the rank of the matrix in terms of the ranks of the blocks? ...
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13 views

How to know function is increasing function for multi variable case?

If there is no leading power from the multi variable function, and if one of the partial derivative is negative and other partial derivative is positive How do I figure out whether the function ...
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26 views

Perturb a piecewise-linear path to make it $C^\infty$

I'm trying to prove that any two points on a path connected smooth manifold can be joined by a smooth path. It becomes easy if I can prove the following: Given a curve $\gamma :\mathbb{R} \to ...
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11 views

Calculate the flux through a closed surface

While studying for a test I have encountered such a task: Calculate the flux through a closed surface, where $S$ is a boundary of area $V$ with an outward orientation. The data: ...
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19 views

Metric tensor for n-sphere in ambient coordinates

Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$: $$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$ What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ ...
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63 views

A triple integral dancing in the unit cube

Straight integration seems pretty tedious and difficult, and I suppose that the symmetry might possibly open some new ways of which I'm not aware. What would your idea be? $$\int_0^1 \int_0^1 ...
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17 views

Showing a multivariable function isn't continuous.

Suppose I wanted to show some multivariable (specifically, 2 variables, is what im referring to) function wasn't continuous. What ways are there to go about doing that? From what I know, there seem to ...
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21 views

Find the critical point and show it is not a global minimizer (using Hessian)

Consider the function $f(x,y) = x^3 + e^{3y}-3xe^y$. Show that $f$ has exactly one critical point and that this point is a local minimizer, but not a global minimizer. I have attempted this, but ...
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17 views

Methods of finding the line of intersection of two planes

I've experienced some confusion regarding the method to finding the line of intersection between two planes. I am aware of the method involving the cross product of the two plane's normal vector ...
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37 views

Find a formula for $f(x, y)$ given the following assumptions…?

I've been going through some examples in my textbook ready for a uni exam in a few days, and I am having difficulty with a few of the questions, in particular this one: A gene is a sequence of ...
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1answer
39 views

For the line $y = mx$, let $m = tan(\theta)$. Write $f(x, mx)$ as a function of $\theta$..?

I have a problem, and I am not sure how to solve it. This is the problem from my book: let $f(x, y)$ be given by the function: $$ f(x, y) = \begin{cases} \frac{2xy}{x^2 + y^2}, & (x, ...
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22 views

Find a smooth path along which a given function on the plane is not differentiable at the origin

From Bamberg & Sternberg’s A Course In Mathematics For Students of Physics, Exercise 6.1d: Let $F(x,y) = \frac{x^3y}{x^2+y^2}$ for $(x,y) \neq (0,0)$ and $F(0,0)=0$. Invent a smooth curve ...
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34 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
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1answer
46 views

Limit of Multi-variable Function

Question What condition must non-negative integers m, n and p satisfy so that $$\lim_{(x,y)\to(0,0)}\frac{x^my^n}{(x^2+y^2)^p}$$ exist? Prove your answer. [Note: if $m=n=p=0$, then the limit ...
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1answer
38 views

Equivalence of Gradient Fields and Exact Differentials on a Non-Simply Connected Region

I've recently been taking 18.02sc Multivariable Calculus on MIT OpenCourseWare, which states the following in one of their course notes: $$M \hat i + N \hat j = \nabla f \implies M dx + N dy \text{ ...
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34 views

Is a circle in the xy plane considered a graph?

So I know a circle is not a function, but is it called a graph? Or can only functions have graphs? Would the circle be better described as a level set of a multivariable function?
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23 views

Proof weird function is discontinuous/has no partial derivatives.

I'm asked to analyze the continuity and existence of partial derivatives at the origin, and even though it seems pretty obvious that this function is discontinuous at that point, I can't seem to prove ...
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2answers
56 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
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1answer
32 views

Chain Rule in Polar coordinates

I was looking for an intuitive explanation for the total derivative in polar coordinates. Let me be somewhat more specific: Take a standard line of reasoning that the gradient w.r.t. polar coordinates ...
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23 views

Finding a minimum of a function, measuring the sum of the squares of distance from some points of the $\mathbb{R}^n$

Given are a finite number of points $a_1, ..., a_m \in \mathbb{R}^n$. Consider the sum of the squares of distance: $$f(x) = \sum_{k=1}^m ||x-a_k||^2, x \in \mathbb{R}^n$$ with $||.||$ being the ...
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32 views

Uniqueness of tangent plane

Let $\Sigma$ be a smooth surface defined as a surface admitting a parametrisation $\boldsymbol{r}:D\subset\mathbb{R}^2\to\mathbb{R}^3$ such that $\boldsymbol{r}$ is of class $C^1(\mathring{D})$ (and ...
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32 views

A little hard double integral

$\iint \frac{2x^2e^{x^2}}{x^2+y^2}dxdy\::\:D=\left\{1\le x\le 2,\:0\le y\le x\right\}$ I use the substitution: $u=x^2,\:v=\frac{y}{x}$ $$$$Then I get: ...
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30 views

The inverse function theorem, problem with the proof

I'm trying to go through proof of an inverse function theorem in multivariable analysis (described in Rudin's handbook) and I'm having problems understanding the part of the proof that deals with $f$ ...
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39 views

Calculating $\iint_{D} \left(x-y\right)dxdy$ where $D=\left\{0\le x-y\le 1,\:1\le xy\le 2\right\}$

$$\iint_{D} \left(x-y\right)dxdy$$ where $D=\left\{0\le x-y\le 1,\:1\le xy\le 2\right\}$ So the substitution is pretty obvious, but j is: $J\:=\frac{1}{x+y}$ $$$$ I dont see how I get rid of the ...
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21 views

Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$

I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative ...
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5 views

Relation between Gâteaux derivatives and partial derivatives

Definition Let $V_1,...,V_n,W$ be nonzero normed spaces over $\mathbb{K}$ and $E$ be open in $ \prod_{i=1}^n V_i$ and $p\in E$. Define $U_i=\{a\in V_i : ...
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1answer
34 views

Taylor Series General Formulas

I'm looking at 2 different Wikipedia pages: The formula here is different than the one given at the end of the section here. Aside from the remainder, why choose one over the other? I'm assuming ...
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34 views

Injectivity of the function $x||x||$ on $\mathbb R^n$

Let , $f:\mathbb R^n\to \mathbb R^n$ be a function defined by $f(x)=x||x||^2$ for $x\in \mathbb R^n$. Then , which are correct ? (A) $f$ is one-one. (B) $f$ has an inverse. Here $f$ is not a ...
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46 views

Why does $\lim_{x\to 0} \frac {\sin (xy)}{x} \to y $?

Let $f(x,y) = \frac{\sin (xy)}{x}$ for $x\neq 0$. How should you define $f(0,y)$ for $y\in \mathbb{R}$ so as to make $f$ a continuous function on all of $\mathbb{R}^2$? So in order for a function to ...
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51 views

Why is $\frac{\partial }{\partial y}\int M dx = \int \frac{\partial M}{\partial y}dx$

$M$ is a function of $x$ and $y$. I'm getting this question from looking at the solution of the exact equation $M \mathrm{dx} + N\mathrm{dy} = 0$.
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53 views

Why is the Lagrange Multipliers Theorem not working?

Consider the function $h: K \to \mathbb{R}$ where $K := \{x \in \mathbb{R}^3:x,y,z \geq 0, x+2y+3z\leq 6\}$. $h$ is defined as: $$ h(x) = xe^{(x+2y+3z)} $$ Find the supremum and the ...
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21 views

Absolute convergence of vector series proof

In Hubbard's multivariable calculus book there is this theorem: If $\sum_{i=1}^{\infty}|\vec a_i|$ converges, then $\sum_{i=1}^{\infty}\vec a_i$ converges. It is said in the book that the ...
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1answer
30 views

Inequality for the gradient of a power of absolute value

Let $U \subset \mathbb{R}^2$ be open, and let $f : U \to \mathbb{C}$ be a smooth complex-valued function which does not vanish anywhere on $U$. Let $r > 0$ be a real constant. Does the ...
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31 views

Graphs of functions and level sets

While going through the first few chapters of my multivariable calculus book, I came across the following: The graph of a function of two variables is a surface in $\mathbb{R}^3$ and is a level ...
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40 views

Spivak's smooth partition of unity [duplicate]

You are right for your link But In your address, There is not any solution for this question and somebody had said that $f$ is redandant without that present even a reason or one proof or a rational ...
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1answer
18 views

Elementary surface integral computation

I'm working on studying for the GRE. I did this problem from Stewart's Calculus, but my answer differs from that in the back of the book. The problem is: Find the area of the part of the sphere ...
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1answer
30 views

Problem about a multivariable calculus

Decide for which of the functions $F:\mathbb R^3\to\mathbb R^3$ given below , there exists a function $f:\mathbb R^3 \to \mathbb R$ such that $(\nabla f)(x)=F(x)$. (A) ...
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1answer
43 views

Limits in multivariable function

$$\lim \limits_{(x, y) \to (0,0)} {x^3 + \sin(x^2+y^2)\over{y^4 + \sin(x^2+y^2)}}$$ I don't visualize a limited function anywhere to evaluate this limit (by the way, I have the information that this ...
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1answer
36 views

Check whether this is indeed a counterexample

Let $A,B \subset \mathbb{R}$; let $Q := A \times B$; and let $f: Q \to \mathbb{R}$ be bounded. The problem is to give a counterexample to the proposition that if the Riemann integral $\int_{Q}f$ ...
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50 views

Find multi-variable function that will make the statements true.

Let x and y denote the concentrations of two proteins encoded by the genes A and B respectively. Let f(x, y) be the rate of change of the concentration of protein A. Find a formula for f(x, y), given ...
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8 views

How to check quasi convexity or quasi concavity using principal minor

Do you check only leading principal minors for verifying Quasi Convexity or Quasi Concavity? Does border of the bordered hessian matrix consist of first derivative of original function if there are ...
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60 views

I need help to solve this function [on hold]

given that $f(x,y,z)=xy^2-y^2+z^2$ solve $$ \frac{\partial}{\partial x} \left( \frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x}\right)=0 $$ ...
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48 views

$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ [on hold]

I need help, I dont understad how it do $ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ please please ...
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3answers
50 views

Proving that a set is open using epsilons.

I'm trying to prove that the set $$A=\{x=(x_{1},x_{2})\in\mathbb{R}^2:x_{1}^{2}+x_{2}^{2}>1\}$$ is open in $\mathbb{R}^2$ with the usual norm is open with the definition of "epsilons". My attempt ...
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19 views

What exactly is the critical point of ln(xy)

Can $\ln(xy)$ be a strictly concave function without a critical point? It seems that the graph has no critical point, therefore there doesn't seem to be maximum point. What does this imply?
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16 views

Understanding the meaning behind plotting a “gradient vector” on a graph containing contour lines?

A rather basic question here, please do forgive any technical errors in the question. Throughout this example consider the general function w=f(x,y) I am used to visualizing derivatives the same way ...
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1answer
22 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
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26 views

Is the total differential the same as the directional derivative?

The way I understand it, the total differential and the directional derivative are both linear approximations of the change in a function at a certain point. So if I know the change in $x$ and $y$ ...
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1answer
29 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
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2answers
20 views

How can I show that the limit of this function under these conditions does not exist?

Show that the limit of the function, $f(x,y)=\frac{xy^2}{x^2+y^4}$, does not exist when $(x,y) \to (0,0)$. I had attempted to prove this by approaching $(0, 0)$ from $y = mx$, assuming $m = -1$ and ...