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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8 views

For the classical diffusion equation ut = r (5ru) (in 3 space dimensions)

fi nd TWO changes of variables which changes the di ffusion constant from 5 to D = 1 for the new coordinate system?
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1answer
16 views

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$.

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$. I am having difficulty in the problem. Please help.
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0answers
17 views

Difficulty solving equation with partial derivatives

So when I was in school I never went past college algebra. But I have encountered a specific equation which I want to understand. At first I thought that an afternoon's focus would be enough to ...
3
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1answer
36 views

Splitting up a double integral

I need to compute the following integral: $$ 2\pi\nu^2\int^a_be^{x^2}\int_{-\infty}^xerfcx(-y)dydx, $$ where $erfcx(x)=e^{x^2}erfc(x)$, $erfc(x)=1 - erf(x)$, and $erf(x)$ is the error function. The ...
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0answers
17 views

Different Partial Derivatives Used in Curl in Proof of Stokes' Theorem

I was reading this proof of Stokes' theorem, which uses two different forms of partial derivatives in its syntax. The function at hand is $P(x, y, f(x, y))$, describing an arbitrary surface in ...
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3answers
44 views

Orthonomal bases and cross products

I want to show that if I have an orthonormal basis of $\mathbb{R}^3$, say $\{\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}\}$, and if $\boldsymbol{u} × \boldsymbol{v} = \boldsymbol{w}$, then we have ...
3
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4answers
86 views

How I can evaluate $\lim_{(x,y) \rightarrow (0,0)} xy(\frac{1+xy}{x^3+y^3})^{1/3}$

I don't have idea how I can evaluate this double limit $$\lim_{(x,y) \rightarrow (0,0)} xy \left(\frac{1+xy}{x^3+y^3} \right) ^{1/3}$$ could you help me please! I try prove that $f$ is continuous: ...
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3answers
66 views

Limit $\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\sin x^2+y^2}$ [on hold]

Find the limit of: $$\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\left(\sin x^2 \right)+y^2}$$ How to find this limit? What is the most straightforward method?
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2answers
30 views

On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
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0answers
23 views

Unable to perform cable theory equation [on hold]

First time posting, let me know if I'm in the wrong board. In my spare time I enjoy reading. I am currently reading Bioelectromagnetism. I stumbled across the cable theory equation for ...
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0answers
16 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 ...
3
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1answer
47 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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1answer
20 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: ...
1
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1answer
36 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 $\mathbb{R}^{n+1}$ ...
5
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1answer
81 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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1answer
18 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 ...
3
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1answer
26 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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1answer
21 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 ...
0
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1answer
46 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 ...
0
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1answer
41 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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0answers
24 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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2answers
35 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
47 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
39 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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2answers
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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2answers
26 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 ...
4
votes
2answers
60 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
34 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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2answers
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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2answers
33 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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0answers
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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0answers
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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1answer
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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1answer
22 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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0answers
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 : ...
2
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1answer
36 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 ...
2
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2answers
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 ...
3
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3answers
47 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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0answers
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 $M \mathrm{dx} + N\mathrm{dy} = 0$.
4
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1answer
54 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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0answers
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 ...
4
votes
2answers
35 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 ...
2
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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
44 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 ...
0
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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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1answer
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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0answers
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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1answer
62 views

I need help to solve this function [closed]

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