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 (1)

52
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 ...
42
votes
3answers
6k 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 ...
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 ...
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 ...
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 ...
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 ...
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)? ...
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 ...
26
votes
5answers
2k 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
4answers
1k 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 ...
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{ ...
23
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 ...
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 ...
19
votes
1answer
506 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
5answers
1k 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 ...
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! :)
16
votes
6answers
388 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 ...
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 ...
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, ...
15
votes
3answers
722 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
1answer
389 views

Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?

I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
14
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 ...
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 ...
14
votes
3answers
511 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
4answers
1k 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 ...
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 ...
12
votes
6answers
363 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
3answers
827 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
2answers
209 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
193 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
183 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 ...
11
votes
3answers
483 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
2answers
326 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 ...
11
votes
2answers
324 views

Integral of determinant

Good evening. I need help with this task $$ \int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi\int\limits_{-\pi}^\pi{\det}^2\begin{Vmatrix}\sin \alpha x&\sin \alpha y&\sin \alpha z\\\sin \beta ...
11
votes
1answer
307 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
1answer
243 views

Find all functions $F(x,y)$ such as $\frac{\sqrt{3}}{2}\frac{\partial{f}}{\partial{x}}+\frac{1}{2}\frac{\partial{f}}{\partial{y}}=0$

How to find all possible functions $f(x,y)$ such as: $$ \frac{\sqrt{3}}{2}f_x+\frac{1}{2}f_y=0$$ (with $f_x = \frac{\partial{f}}{\partial{x}}$ ) Here's everything I tried: 1) I can guess the ...
11
votes
1answer
435 views

Problem 3-38 in Spivak´s Calculus on Manifolds

This is not homework. Problem 3-38 reads: Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
11
votes
2answers
114 views

$\int_{\mathbb{R}}|f(t)|^2dt=\int_{\mathbb{R}}|f'(t)|^2dt$ implies $f(t)=\mathbb{x}_{i}|f(t)|$

Let $f \in C^{1}(\mathbb{R},\mathbb{R}^m)$ be such that $f$ and $f'$ are square integrable and $$\{t:f(t)=0\} \subset \{t:f'(t)=0\}$$ $$ |\{t:f(t)=0\}|=n\in \mathbb{N}$$ Prove that if ...
10
votes
3answers
352 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 ...
10
votes
2answers
1k 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 ...
10
votes
2answers
353 views

Curl(curl(A)) with Einstein Summation Notation

I have two questions on the computation of $\nabla \times (\nabla \times \mathbf{A}) $ with Einstein summation notation based on http://www.physics.ohio-state.edu/~ntg/263/handouts/tensor_intro.pdf. ...
10
votes
3answers
793 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 $U$ ...
10
votes
2answers
833 views

Are “differential forms” an algebraic approach to multivariable calculus?

I am recently learning some basic differential geometry. As I understand, differential forms provide a neat way to deal with the topics in calculus such as Stoke's theorem. In order to define the ...
10
votes
2answers
91 views

The “second derivative test” for $f(x,y)$

I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...
10
votes
3answers
702 views

Why is $(0, 0)$ not a minimum of $f(x, y) = (y-3x^2)(y-x^2)$?

There is an exercise in my lists about those functions: $$f(x, y) = (y-3x^2)(y-x^2) = 3 x^4-4 x^2 y+y^2$$ $$g(t) = f(vt) = f(at, bt); a, b \in \mathbf{R}$$ It asks me to prove that $t = 0$ is a ...
10
votes
1answer
415 views

Visualizing Commutator of Two Vector Fields

I'm reading a book on calculus, the part about vector fields on manifolds. It's a nice book, but with a severe drawback --- it has no pictures. I like how vectors are treated algebraically, as ...
10
votes
3answers
296 views

Determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists.

I am trying to determine whether $\lim_{(x,y)\to (2,-2)} \dfrac{\sin(x+y)}{x+y}$ exists. I should be able to use the following definition for a limit of a function of two variables: Let $f$ be a ...
10
votes
1answer
1k views

Curl of a vector in spherical coordinates

The curl of a Vector function in curvilinear coordinate system is given by $$ \nabla \times A = \frac 1 {h_1 h_2 h_3} \begin{vmatrix} h_1 \hat e_1 & h_2 \hat e_2 & h_3 \hat e_3\\ \partial ...
10
votes
5answers
242 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 ...
10
votes
1answer
322 views

Writing Integrals using Differential Forms

Consider some smooth curve $C \subset \mathbb{R^n}$ and $\gamma:[a,b] \subset\mathbb{R}\rightarrow C$ a parametrisation of $C$ and a continuous vector field $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...