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 ...
31
votes
5answers
5k 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 ...
15
votes
8answers
4k views

References for the multivariable 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
1k 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 ...
29
votes
5answers
4k 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)? ...
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?
19
votes
5answers
24k 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
862 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
332 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
284 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 ...
30
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 ...
16
votes
4answers
3k 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, ...
4
votes
1answer
416 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
566 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
378 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
784 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 ...
26
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{ ...
20
votes
2answers
3k 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 ...
17
votes
6answers
5k 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. ...
4
votes
5answers
781 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 ...
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 ...
2
votes
1answer
863 views

Proof on showing if F(x,y,z)=0 then product of partial derivatives (evaluated at an assigned coordinate) is -1

The task is as follows: Given: $$F(x,y,z) = 0$$ Goal: Show $\frac{\partial z}{\partial y}$ (evaluated at $x$) * $\frac{\partial y}{\partial x}$ (evaluated at $z$) * $\frac{\partial ...
12
votes
6answers
426 views

Orientation of Boundary of Lower Hemisphere - Stokes's Theorem

I don't understand how to arrive at the red part in the given solution for question #19 from Section 16.8, Calculus, 6th Ed, by James Stewart. I've read Determining correct orientation for ...
7
votes
2answers
226 views

Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$ - 2013 10C

2013 10C. Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable ...
3
votes
3answers
136 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
300 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 ...
12
votes
3answers
988 views

Equation of Cone vs Elliptic Paraboloid

I can't understand why $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{*}$$ corresponds to an elliptic paraboloid and $$\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2} \tag{**}$$ to a cone, ...
11
votes
4answers
289 views

Which Cross Product for the Desired Orientation of a Sphere ? [Stewart P1091 16.7.23]

P1086: For a closed surface, the positive orientation is the one for which the normal vectors point outward from the surface, and inward-pointing normals give the negative orientation. P1087: ...
3
votes
2answers
93 views

Proof Strategy - Without Stokes's Theorem - Calculate $\iint_S \operatorname{curl} \mathbf{F} \cdot\; d\mathbf{S}$ for $\mathbf{F} = yz^2\mathbf{i}$

2013 10C. Question: Consider the bounded surface S that is the union of $x^2 + y^2 = 4$ for $−2 \le z \le 2$ and $(4 − z)^2 = x^2 + y^2 $ for $2 \le z \le 4.$ Sketch the surface. Use suitable ...
3
votes
2answers
87 views

$\theta$ for Triple Integral above paraboloid $z = x^2 + y^2$ and below $z = 2y$ [Stewart P1011 15.8.37]

$\bf\sf37.$ Evaluate $\iiint_E z\,dV,$ where $E$ lies above the paraboloid $z=x^2+y^2$ and below the plane $z=2y.$ In cylindrical coordinates the paraboloid is given by $z=r^2$ and the plane by ...
3
votes
2answers
734 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
2answers
92 views

Question about Implicit function theorem

I was asked a simple question, show that $y+\sin y=x$ sets in the neighborhood of $(0,0)$ $y$ as a function of $x$, and find $\dfrac{dy}{dx}(0,0)$ Firstly, my naive solution would be: Since $lim_{y ...
2
votes
1answer
186 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
179 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
2answers
417 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 ...
1
vote
3answers
217 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
59 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 ...
12
votes
4answers
239 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 ...
12
votes
1answer
979 views

Nice way of thinking about the Laplace operator… but what's the proof?

Here's a nice fact: roughly speaking, the Laplace operator gives you the difference between the value of a function at a point and the average value at "neighboring" points. More precisely, in ...
8
votes
4answers
14k 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 ...
9
votes
1answer
469 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
742 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 ...
6
votes
0answers
193 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 ...
3
votes
2answers
436 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 ...