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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47
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5answers
12k views

(Theoretical) Multivariable Calculus Textbooks [duplicate]

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope ...
37
votes
4answers
5k views

Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$

It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why? I got: $2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\...
12
votes
2answers
4k views

Anti-curl operator

It is known that if a vector field $\vec{B} \;\;\;$is divergence free, and defined on $ R^3 $ then it can be shown as $\vec{B}=\nabla\times\vec{A} \;\;$ for some vector field A. Is there a way to ...
25
votes
6answers
6k views

Limit is found using polar coordinates but it is not supposed to exist.

Consider the following 2-variable function: $$f(x,y) = \frac{x^2y}{x^4+y^2}$$ I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$. I used polar coordinates instead of ...
12
votes
0answers
2k views

Volume of an n-simplex [duplicate]

It's rather tedious to show using Fubini's Theorem and induction on $n$ that the volume of the region $x_1+x_2+...+x_n \leq 1$ with $x_1,...,x_n$ nonnegative is $\frac{1}{n!}$. Is there an easier way ...
60
votes
6answers
9k views

Intuitive interpretation of the Laplacian

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian (a.k.a. divergence of gradient)?
42
votes
8answers
53k views

What is Jacobian Matrix?

What is Jacobian matrix? What are its applications? What is its physical and geometrical meaning? Can someone explain with examples? Thanks! :)
3
votes
6answers
1k views

Prove that $\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$

$$\lim_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$$ Please, Anyone could suggest me some way for this?. Thanks.
39
votes
9answers
13k views

References for the multivariable calculus

Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing ...
29
votes
2answers
10k views

Is there a step by step checklist to check if a multivariable limit exists and find its value?

Do we rely on certain intuition or is there an unofficial general crude checklist I should follow? I had a friend telling me that if the sum of the powers on the numerator is smaller then the ...
8
votes
3answers
1k views

Why isn't there a continuously differentiable injection into a lower dimensional space?

How to show that a continuously differentiable function $f:\mathbb{R}^{n}\to \mathbb{R}^m$ can't be a 1-1 when $n>m$? This is an exercise in Spivak's "Calculus on manifolds". I can solve the ...
5
votes
3answers
2k views

How do I differentiate this integral?

That is: $$\left(\int_{a(x)}^{b(x)}\!f(x,t)\,dt\right)'$$ I don't know how to differentiate a integral if functions of $x$ are at its limits. Can you guys show me how to do this?
3
votes
4answers
1k views

Multivariable Calculus Book Reference

I am looking for a multivariable calculus book that is really physics oriented. Anyone know of any? EDIT: My wife is looking to brush up on multivariable at the same time she needs to brush up on ...
6
votes
5answers
5k views

How to calculate gradient of $x^TAx$

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior ...
5
votes
1answer
758 views

Partial derivative confusion.

I don't understand partial derivatives. Here's an example that nails down my confusion: Suppose we have some variables $x$, $p$, and $q$ with $p=x^2$ and $q=e^x$. Then $$\frac{\partial q}{\partial p} ...
11
votes
5answers
464 views

Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 +dz^...
4
votes
1answer
172 views

A differentiation under the integral sign

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function Lebesgue summable on all $\mu$-measurable and bounded subsets of $\mathbb{R}^n$, where $\mu$ is the usual Lebesgue measure defined on $\mathbb{R}^n$, ...
3
votes
3answers
458 views

How to solve this integral for a hyperbolic bowl?

$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
2
votes
3answers
95 views

Showing that $\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$

Show that $$\lim_{(x,y) \to (0, 0)}\frac{xy^2}{x^2+y^2} = 0$$ I have tried switching to polar coordinates but I'm not getting a single term. This is what I did. Putting $$x=r\sin θ,\quad y=r\cos ...
34
votes
4answers
3k views

Volume of Region in 5D Space

I need to find the volume of the region defined by $$\begin{align*} a^2+b^2+c^2+d^2&\leq1,\\ a^2+b^2+c^2+e^2&\leq1,\\ a^2+b^2+d^2+e^2&\leq1,\\ a^2+c^2+d^2+e^2&\leq1 &\text{ ...
33
votes
2answers
3k views

Asymmetric Hessian matrix

Are there any functions, $f:U\subset \mathbb{R}^n \to \mathbb{R}$, with Hessian matrix which is asymmetric on a large set (say with positive measure)? I'm familiar with examples of functions with ...
3
votes
2answers
1k views

An application of partitions of unity: integrating over open sets.

In Spivak's "Calculus on Manifolds", Spivak first defines integration over rectangles, then bounded Jordan-measurable sets (for functions whose discontinuities form a Lebesgue null set). He then uses ...
5
votes
3answers
257 views

Does the limit $\lim_{(x,y)\to (0,0)} \frac {x^3y^2}{x^4+y^6}$ exist

$$ \lim_{(x,y)\to (0,0)} \frac {x^3y^2}{x^4+y^6} $$ Does this limit exist? I've tried substituting y=x^0.5 and y=x^(2/3) which both goes to 0.
4
votes
3answers
3k views

A continuously differentiable map is locally Lipschitz

Let $f:\mathbb R^d \to \mathbb R^m$ be a map of class $C^1$. That is, $f$ is continuous and its derivative exists and is also continuous. Why is $f$ locally Lipschitz? Remark Such $f$ will not be ...
3
votes
2answers
2k views

What is the difference between $d$ and $\partial$?

After seeing the following equation in a lecture about tensor analysis, I became confused. $$ \frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds} $$ What exactly is the difference ...
3
votes
2answers
2k views

Dimensions of a box of maximum volume inside an ellipsoid

Finding the dimensions of the maximum volume box inside the ellipsoid. I assume that the volume of a box, $V(x,y,z) = xyz$ (they did not give this to me, but this is the volume of a box right?) ...
2
votes
1answer
1k views

Mass of a rectangle

Let $a,b,c$ be positive real numbers such that $c<a$. Suppose given is a thin plate $R$ in the plane bounded by $$\frac{x}{a}+\frac{y}{b}=1, \frac{x}{c}+\frac{y}{b}=1, y=0$$ and such that the ...
41
votes
2answers
2k views

Integration of forms and integration on a measure space

In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus): the ...
28
votes
5answers
4k views

What is an intuitive explanation for $\operatorname{div} \operatorname{curl} F = 0$?

I am aware of an intuitive explanation for $\operatorname{curl} \operatorname{grad} F = 0$ (a block placed on a mountainous frictionless surface will slide to lower ground without spinning), and was ...
26
votes
5answers
12k views

What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
19
votes
2answers
505 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
13
votes
3answers
2k views

Connectivity, Path Connectivity and Differentiability

I have two questions which pertain to differentiability, connectivity and path connectivity. Ocasionally, I will encounter an author who defines connectivity in the following way: An open subset $U$ ...
14
votes
4answers
334 views

How prove this integral limit $=f(\frac{1}{2})$

Let $f$ be a continuous function on the unit interval $[0,1]$. Show that $$\lim_{n\to\infty}\int_{0}^{1}\cdots\int_0^1\int_{0}^{1}f\left(\dfrac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)dx_{1}dx_{2}\cdots ...
11
votes
4answers
652 views

Volume of $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$

Let $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$. I know $T_n$ is tetrahedron. My question: How can I compute the volume of $T_n$ for every $n$?
8
votes
3answers
3k views

Property of Dirac delta function in $\mathbb{R}^n$

How does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$ where $S$ is the surface inside $V$ where $g(\...
20
votes
2answers
2k views

What is the intuition behind the Wirtinger derivatives?

The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I ...
4
votes
1answer
3k views

Does the Divergence Theorem Work on a Surface?

The divergence theorem in $\mathbb{R}^3$ says that the integral of the divergence of a vector field over a solid $\Omega$ in $\mathbb{R}^3$ equals the flux through the surface of $\Omega$ denoted by $\...
3
votes
4answers
894 views

Computing A Multivariable Limit: $\lim_{(x,y) \to (0,0)}\frac{2x^2y}{x^4 + y^2}.$

Please help me in computing the following limit. $$\lim_{(x,y) \to (0,0)}\frac{2x^2y}{x^4 + y^2}.$$ This will be my first attempt in computing a limit involving 2 variables. Is this a part of ...
3
votes
1answer
2k views

What is the meaning of evaluating the divergence at a _point_?

Reading this first, Divergence (div) is “flux density”—the amount of flux entering or leaving a point. Think of it as the rate of flux expansion (positive divergence) or flux contraction (negative ...
3
votes
1answer
4k views

surface area of torus of revolution

Here's a question from one of my exercises, Exercise 14. Let $C$ be a curve in $\mathbb{R}^2$ given by parametric equations $x=f(t)$, $y=g(t)$. Let $S$ be the surface of revolution of the curve $C$...
0
votes
1answer
138 views

The Function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ defined by $f(r,\theta)=(r\cos\theta,r\sin\theta)$

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be the function defined by $f(r,\theta)=(r\cos\theta,r\sin\theta).$ Then for which of the open subset $U$ of $\mathbb{R}^2$ given below, $f$ restricted ...
7
votes
3answers
217 views

Derivative of function with 2 variables

I've leart in Calculus 1 that the derivetive of $f(x)$ is: $$\lim_{h\to0} \frac{f(x+h) - f(x)}{h}$$. suppose $f(x,y)$ is a function with 2 variables, does $$f'(x,y) = \lim_{h\to0} \frac{f(x+h, y+h) ...
3
votes
1answer
102 views

Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$

Let $f:\mathbb R^n\to\mathbb R$ differntiable and $f(0)=0$. Prove exist $g_i$ s.t for $x=(x_1,\dots,x_n):f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$. hint:$f(x)=\int\limits_0^1f\prime(tx)dt$. I dont ...
2
votes
1answer
295 views

Is this definition missing some assumptions?

In his "Calculus on manifolds" Spivak first defines $n$-dimensional (Riemann-) integral over rectancles, then over Jordan measurable subsets of rectangles and finally extends it to open sets using ...
1
vote
1answer
229 views

Uniform convergence in a proof of a property of mollifiers in Evans's Partial Differential Equations [duplicate]

Here are some definitions that was taken of PDE Evans book: Here is a proof of a property of mollifiers: My (elementary) question is: Why is the convergence uniform on $V$? Thanks.
5
votes
2answers
319 views

Multivariable Delta Epsilon Proof $\lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4}$ — looking for a hint

I have the limit $$\lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4},$$ and would like to show with an $\epsilon-\delta$ proof that it is zero. I know with a situation like $$\left|\frac{x^4y}{x^4+y^4}\right|...
4
votes
1answer
630 views

How does one prove if a multivariate function is constant?

Suppose we are given a function $f(x_{1}, x_{2})$. Does showing that $\frac{\partial f}{\partial x_{i}} = 0$ for $i = 1, 2$ imply that $f$ is a constant? Does this hold if we have $n$ variables ...
2
votes
1answer
379 views

The implication of zero mixed partial derivatives for multivariate function's minimization

Suppose $f(\textbf x)=f(x_1,x_2) $ has mixed partial derivatives $f''_{12}=f''_{21}=0$, so can I say: there exist $f_1(x_1)$ and $f_2(x_2)$ such that $\min_{\textbf x} f(\textbf x)\equiv \min_{x_1}...
2
votes
5answers
228 views

Implicit differentiation

I want to differentiate $x^2 + y^2=1$ with respect to $x$. The answer is $2x +2yy' = 0$. Can some explain what is implicit differentiation and from where did $y'$ appear ? I can understand that $...
2
votes
4answers
937 views

Using Spherical coordinates find the volume:

Inside the surfaces $z=x^2+y^2$ and $z=\sqrt{2-x^2-y^2}$ I integrated over the ranges: $0 \leq \theta \leq 2\pi$ $ 0 \leq \phi \leq \frac{\pi}{2}$ $0 \leq r \leq \sqrt{2}$ I get $\frac{\pi}{2}(4\...