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 ...
32
votes
5answers
6k 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 ...
17
votes
9answers
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 ...
9
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 ...
2
votes
7answers
253 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.
32
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?
20
votes
5answers
25k 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
892 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
334 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
295 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 ...
31
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 ...
12
votes
4answers
264 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 ...
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
434 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
578 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
400 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
789 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 $N(t)$. $T(t) =$ $N(t) =$ ...
42
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{ ...
21
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
3k 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
913 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
908 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
435 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
227 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
139 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
1k 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
294 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
95 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
92 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
346 views

Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables

Problem 3-29 (p. 61) in the section treating Fubini´s theorem reads: Use Fubini´s theorem to derive an expression for the volume of a set of $\mathbb{R}^{3}$ obtained by revolving a Jordan-measurable ...
3
votes
2answers
742 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
100 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
4answers
2k views

Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$

"Find the shortest distance between the point $(8,3,2)$ and the line through the points $(1,2,1)$ and $(0,4,0)$" $$P = (1,2,1), Q = (0,4,0), A = (8,3,2)$$ $OP$ = vector to $P$ $$PQ_ = (0,4,0) - ...
2
votes
1answer
189 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
2answers
144 views

Show that $g$ is continuous at $(0,0)$.

Let $g:\mathbb{R^2}\rightarrow \mathbb{R} $ so that, in $M=[0,1]\times[0,1]$, $$g(x,y)=\begin{cases}\frac{x^2+y^2}{x+y} &\text{ if }x+y \neq 0,\\\\ 0&\text{ if }x+y=0\end{cases}$$ Show that ...
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
448 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 ...
0
votes
1answer
633 views

Find a plane that passes through a point and is perpendicular to 2 planes

Find an equation of a plane that passes through $p(1,5,1)$ and is perpendicular to planes $2x+y-2z = 2$ and $x+3z=4$. I basically need the 2 other points to make the vector and perform the cross ...
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
1answer
1k 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 ...