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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26
votes
4answers
3k 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)\mathrm ...
29
votes
5answers
4k 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 ...
14
votes
8answers
3k views

References for the multivariate calculus

Maybe 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 ...
8
votes
2answers
994 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 ...
5
votes
3answers
1k 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?
17
votes
5answers
22k 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! :)
10
votes
3answers
794 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 $U$ ...
3
votes
3answers
302 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
1answer
276 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 ...
30
votes
2answers
1k 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 ...
27
votes
5answers
3k 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)? ...
28
votes
2answers
2k 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 ...
4
votes
1answer
394 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} ...
3
votes
2answers
528 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 ...
2
votes
4answers
317 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 ...
2
votes
1answer
2k 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 ...
0
votes
1answer
749 views

Unit Tangent and Unit Normal Vectors — Calculus III Question

Consider the following vector function. $$r(t) = \left\langle 2t \cdot \sqrt{2}, e^{2t}, e^{-2t}\right\rangle$$ (a) Find the unit tangent and unit normal vectors $T(t)$ and $N(t)$. $T(t) =$ $N(t) =$ ...
41
votes
1answer
2k views

What is the solution to Nash's problem presented in “A Beautiful Mind”?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
23
votes
4answers
1k 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{ ...
15
votes
5answers
4k 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 ...
7
votes
3answers
1k 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 ...
6
votes
5answers
2k views

Need Help: Any good textbook in undergrad multi-variable analysis/calculus?

This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: -Differentiability. -Open mapping theorem. ...
3
votes
3answers
128 views

If $\frac{\partial \varphi}{\partial x}=f(x,y),\frac{\partial\varphi}{\partial y}=g(x,y)$, what is $\varphi$?

Suppose we have a real-valued function $\varphi(x,y)$ such that $$ \frac{\partial \varphi}{\partial x}=f(x,y)\quad\text{and}\quad\frac{\partial\varphi}{\partial y}=g(x,y) $$ for some functions $f$ and ...
3
votes
2answers
296 views

Calculus of an integral

I'm trying to calculate the following integral $$\int\limits_S \exp\left\{\sum_{i=1}^n \lambda _ix_i\right\} \, d\sigma$$ where the $\lambda_i$ are constant real parameters, $S$ is a surface in ...
3
votes
2answers
684 views

Scalar Product for Vector Space of Monomial Symmetric Functions

Suppose a multinomial $P(X_1, X_2,\ldots, X_n)$, that is given as a sum of monomials $m_\lambda$ with coefficients $c_k$: $$ P(\vec{X})=P(X_1, X_2,\ldots, X_n) = \sum_k c_k m_{\lambda_k} . $$ Since ...
2
votes
1answer
176 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
0answers
178 views

Does this hold?

Strayed on the following question. Assume that $x_{1}$,$\ldots$, $x_{d}\ge0$ with $x_{1}+\ldots+x_{d}=1$ and $y_{1},\ldots,y_{d}\in\mathbb{R}$. Does $$ \min_{1\le i\ne j\le ...
1
vote
3answers
196 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.
0
votes
2answers
116 views

How does partial derivative work?

I don't understand the second step at all. Where did the $\partial^2 u/ \partial x^2$ come from and why do we have six terms?
-5
votes
1answer
58 views

Multivariable calc. [closed]

Show that for $\vec{f}(t)=(\cos t,\sin t,t)$, the Mean Value Theorem does not hold.That is, show that there is no $t$ in the interval $(0,2\pi)$ such that ...
19
votes
2answers
2k 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
12k views

Surface Element in Spherical Coordinates

In spherical polars, $$x=r\cos(\phi)\sin(\theta)$$ $$y=r\sin(\phi)\sin(\theta)$$ $$z=r\cos(\theta)$$ I want to work out an integral over the surface of a sphere - ie $r$ constant. I'm able to derive ...
8
votes
2answers
101 views

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

let $f$ be a continuous function on the uint inteval $[0,1]$ ,Show that $$\lim_{n\to\infty}\int_{0}^{1}\cdots\cdots\int_{0}^{1}f\left(\dfrac{x_{1}+x_{2}+\cdots+x_{n}}{n}\right)dx_{1}dx_{2}\cdots ...
9
votes
1answer
443 views

Information captured by differential forms

My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ...
8
votes
1answer
702 views

If derivative of a function is the zero function in $\mathbb R^n$, then the function is constant when the domain is path-connected

Some definitions first. Let $A \subseteq \mathbb R^n$. Let $x,y \in A$. A path between $x$ and $y$ is a continuous function $f: [0,1] \rightarrow \mathbb{R}^n$ with $f(0) = x$ and $f(1) = y$. The set ...
5
votes
0answers
182 views

Multivariable Integral, How to compute it?

Can anybody please tell me, how to evaluate a multivariate integral with a gaussian weight function. $$ \mathcal{Z_{n}} \equiv\int_{-\infty}^{\infty} \exp\left(-a\sum_{j = 1}^{n}x_{j}^2\right)\, {\rm ...
15
votes
4answers
2k views

Bridging any “gaps” between AP Calculus and College/Univ level Calculus II

I've been asked to tutor a soon-to-be college freshman who has taken AP Calculus and successfully earned college credit for first semester calculus. He has been admitted to an Engineering program, ...
3
votes
2answers
400 views

A curve parametrized by arc length

Let $C$ be a plane curve parametrized by arc length by $\alpha(s)$, $T(s)$ (unit tangent vector) and $N(s)$ (unit normal vector). Prove that $$\frac{d}{ds} N(s)=-\kappa(s)T(s).$$ I know that ...
2
votes
1answer
3k views

How to tell if a limit of a multi-variable function exists?

Since I began studying limits of multi-variable functions, I have been baffled with this question: how can one tells if a limit exists or not? I don't know if it's the right way to solve this kind of ...
8
votes
5answers
548 views

Distinction between vectors and points

I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ ...
8
votes
2answers
865 views

Can “being differentiable” imply “having continuous partial derivatives”?

Consider the following theorem: Let $E$ be a subset of ${\bf R}^n$, $f:E\to {\bf R}^m$ be a function, $F$ be a subset of $E$, and $x_0$ be an interior point of $F$. If all the partial derivatives ...
6
votes
2answers
1k views

Equivalent condition for differentiability on partial derivatives

I want to extend the concept of derivative of a real function of real variable to a function $f:A\subset \mathbb{R}^n \to \mathbb{R}^m$ with $A$ open. If $x_0 \in A$ then I say that $f$ has derivative ...
5
votes
1answer
376 views

Use Stokes's Theorem to show $\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2$

I am a little stuck on the following problem: Use Stokes's Theorem to show that $$\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2,$$ where $C$ is the suitably oriented intersection of the ...
5
votes
5answers
850 views

Reference for multivariable calculus

I'm looking for a book to learn multivariable calculus that is rigorous, but not overly technical, and also provides meaningful insight. Standard calculus texts like Stewart and Thomas are too ...
4
votes
1answer
195 views

Limit with integral or is this function continuous?

Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center ...
4
votes
3answers
974 views

Definition of the gradient for non-Cartesian coordinates

The gradient of a function $f: \mathbb{R}^n \to \mathbb{R}$ is defined as the vector of the partial derivatives: $$ \nabla f = \left(\frac{\partial f}{\partial x_1}, ..., \frac{\partial f}{\partial ...
2
votes
5answers
474 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 ...
7
votes
1answer
135 views

The proof of the Helmholtz decomposition theorem through Neumann boundary value problem

As you know, the Helmholtz decomposition theorem is as follows. Let $\Omega$ be open and simply connected, and bounded region in $\mathbb R^3$. Then the smooth vector field $E : \Omega \to ...
6
votes
3answers
1k views

L'Hôpital in several variables

I am wondering if there is a multidimensional analog of l'Hôpital's rule for functions of several variables. I have searched online for a while and have found people that argue both sides. One said ...
3
votes
2answers
95 views

Is this vector-valued map Hölder-continuous?

Pick $0<q<1$ and consider the map from $\mathbb{R}^n$ to $\mathbb{R}^n$ that sends $x$ to $|x|^{q-1}x$. Is this map Hölder-continuous (I guess with exponent $\leq q$)? In dimension one, I can ...