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).

learn more… | top users | synonyms

56
votes
1answer
2k views

What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
44
votes
1answer
3k 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 ...
43
votes
3answers
8k views

What is the importance of Calculus in today's Mathematics?

For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a ...
35
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 ...
35
votes
2answers
905 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
34
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)? ...
33
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 ...
32
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 ...
28
votes
4answers
4k 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 ...
28
votes
5answers
2k views

The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
26
votes
4answers
2k 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{ ...
26
votes
5answers
3k 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 ...
25
votes
3answers
1k views

When did Fubini's name get applied to the theorem without measures?

Fubini's theorem, from 1907, expresses integration with respect to a product measure in terms of iterated integrals. The simpler version of this theorem for multiple Riemann integrals was used long ...
22
votes
2answers
4k 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 ...
21
votes
5answers
28k 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! :)
20
votes
9answers
6k 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 ...
19
votes
5answers
2k 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 ...
19
votes
1answer
543 views

How prove there exists a point $(x_{0},y_{0})$, such $\Delta f|_{(x_{0},y_{0})}\ge 0$

Question: Assume that the function $f(x,y)$ is twice continuously differentiable on $\mathbb R^2$, and $$f\big|_{\partial\Sigma}=0,\quad \text{where}\,\,\,\partial\Sigma=\{(x,y)\in\mathbb ...
18
votes
3answers
1k views

Interesting implicit surfaces in $\mathbb{R}^3$

I have just written a small program in C++ and OpenGl to plot implicit surfaces in $\mathbb{R}^3$ for a Graphical Computing class and now I'm in need of more interesting surfaces to implement! Some ...
18
votes
5answers
398 views

Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
17
votes
6answers
6k 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 ...
17
votes
4answers
4k 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, ...
17
votes
4answers
663 views

How to explain this quirk of the chain rule?

Assume I have a function $f = f(y, \phi(y,x))$ and I want to calculate $\frac{\partial f}{\partial y}$, I use the chain rule to get \begin{equation} \frac{\partial f}{\partial y} = \frac{\partial ...
16
votes
6answers
487 views

Why is boundary information so significant? — Stokes's theorem

Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
16
votes
4answers
2k views

Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$

Are there any geometric interpretation for the second partial derivative? i.e. $$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$ In particular, I'm trying to understand the determinant from ...
15
votes
3answers
782 views

Maximizing a sum of inner products

Someone asked this question on a French maths forum here and it caught my attention. The question is the following: let $(E, \langle \cdot, \cdot \rangle)$ be a Euclidean vector space. Find the ...
15
votes
3answers
369 views

Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
14
votes
3answers
528 views

Show that a definite integral vanishes for all values of $a,b,c > 0$.

Show $$\int_0^\infty \int_0^\infty \frac{(ax-by) {\rm e}^{-x} {\rm e}^{-y}}{(a^2 x + b^2 y + c x y)^{\frac{3}{2}}} \,dx \,dy = 0$$ for any $a,b,c > 0$. I came upon the above double integral when ...
14
votes
5answers
279 views

Geometrical Interpretation of Cauchy Riemann equations?

Differentiation has an obvious geometric interpretation, and the Cauchy Riemann equations are closely linked with differentiation. Do the Cauchy Riemann equations have a geometric interpretation?
14
votes
2answers
368 views

How to find the integral of implicitly defined function?

Let $a$ and $b$ be real numbers such that $ 0<a<b$. The decreasing continuous function $y:[0,1] \to [0,1]$ is implicitly defined by the equation $y^a-y^b=x^a-x^b.$ Prove $$\int_0^1 \frac {\ln ...
14
votes
3answers
441 views

$\nabla \cdot \color{green}{(\mathbf{F} {\times} \mathbf{G})} $ with Einstein Summation Notation [Stewart P1068 16.5.27]

$\nabla \cdot \color{green}{(\mathbf{F} {\times} \mathbf{G})} = \partial_h\color{green}{\epsilon_{hij}F_iG_j}$ $ = \epsilon_{hij}\partial_h[F_iG_j]$ $ = \color{purple}{\epsilon_{hij}G_j\partial_hF_i} ...
13
votes
4answers
288 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 ...
13
votes
2answers
2k views

Derivation of the method of Lagrange multipliers?

I've always used the method of Lagrange multipliers with blind confidence that it will give the correct results when optimizing problems with constraints. But I would like to know if anyone can ...
13
votes
4answers
2k views

Why does dust gather in corners?

I've noticed when sweeping the floor that dust gathers particularly in the corners. I assume there is a fluid mechanics reason for this. Does anyone know what it is? Edit: No, really, this is a ...
13
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 ...
13
votes
2answers
285 views

How do I generalize the derivatives / integrals from multivariable calc?

$\newcommand{\RR}{\mathbb{R}}$ This is a long post, so I'll put the big question right at the top: There's a whole lot of derivative-like and integral-like operations. Are they special cases of some ...
12
votes
6answers
533 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 ...
12
votes
5answers
1k views

Evaluating Double Integral $\int_0^b\int_0^x\sqrt{a^2-x^2-y^2}dy\,dx$

What is the best method for evaluating the following double integral? $$ \int_{0}^{b}\int_{0}^{x}\,\sqrt{\,a^{2} - x^{2} - y^{2}\,}\,\,{\rm d}y\,{\rm d}x\,, \qquad a > \sqrt{\,2\,}\,\,b $$ Is ...
12
votes
10answers
420 views

Why is it that $\int_a^b \int_c^d f(x)g(y)\,dy\,dx=\int_a^b f(x)\,dx \int_c^d g(y)\,dy$?

The title sums it up. It's simple to prove, but I'm wondering if there is a geometric interpretation?
12
votes
3answers
249 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
12
votes
2answers
68 views

Frobenius theorem for 2-plane fields on some open set in $\mathbb{R}^3$

I need help with this two part question. I am rather confused by it. let $f(x, y, z)$, $g(x, y, z)$ be smooth on $U \subset \mathbb{R}^3$ with $f^2 + g^2 > 0$ on $U$. Define the differential form ...
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, ...
12
votes
1answer
239 views

Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ...
12
votes
2answers
287 views

Calculating $\zeta(4)$ using iterated integrals

This question concerns the following integral. $$\int_0^1 \int_0^1 \int_0^1 \int_{0}^{1}\frac{dw\,dx\,dy\,dz}{1-wxyz}=\zeta(4)=\frac{\pi^4}{90}.$$ In an answer below (given before this edit), ...
12
votes
1answer
1k 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 ...
12
votes
1answer
377 views

Evaluating a double integral by using $\int\arctan (1+\sin 2t)\ dt/(1+\sin 2t)$

I am trying to evaluate the following integral: $$ I:=\iint_D\frac{\mathrm dy\ \mathrm dx}{1+(x+y)^4} $$ where $D := \{(x,y)\mid x^2+y^2\le1,x\ge0,y\ge0\}$. I tried to evaluate it in the following ...
11
votes
3answers
634 views

What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their ...
11
votes
2answers
212 views

quadratic form corresponding to function at critical point is positive definite implies local minimum

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^3$ function. Have $x_0$ be a critical point of $f$. How would I go about proving that if the quadratic form $q(h)$ corresponding to $f$ at $x_0$ is ...
11
votes
3answers
666 views

Can the curl operator be generalized to non-3D?

In three dimensions, the curl operator $\newcommand{curl}{\operatorname{curl}}\curl = \vec\nabla\times$ fulfils the equations $$\curl^2 = ...
11
votes
4answers
553 views

Double integral over a region

Given $f(x,y)=\displaystyle\frac{x^2}{x^2+y^2}$ and $D=\{(x,y) : 0 \leq x \leq 1, x^2 \leq y \leq 2-x^2\}$ i have to solve $\displaystyle\int\displaystyle\int_Df(x,y)dA$. Here's my try: (1) Changing ...