Tagged Questions

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).

1answer
3k views

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

Let $f$ be a continuous, even 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 $(0,a)$. [You might imagine ...
6answers
9k 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)?
1answer
6k 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 (...
3answers
12k 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 ...
5answers
12k 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 ...
2answers
1k views

4answers
2k views

Is there a vector field that is equal to its own curl?

I was wondering if there is a vector field that satisfies the following condition: $$\vec F=\nabla \times \vec F$$
2answers
3k 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 ...
5answers
32k views

1answer
402 views

Find $f(x,y )$ such that $f_{x},f_{y},f_{yx}$ are continuous,but $f_{xy}$ is not

Let $f$ be a function of two variables,let$(a,b)$ be a point and let $D$ be an open disk with center $(a,b)$. Assume that $f$ is $\mathcal C^{1}$ on $D$, and $f_{yx}$ exist on $D$. Further, the ...
5answers
2k 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 ...
3answers
7k views

Divergence as transpose of gradient?

In his online lectures on Computational Science, Prof. Gilbert Strang often interprets divergence as the "transpose" of the gradient, for example here (at 32:30), however he does not explain the ...
3answers
920 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 ...
2answers
426 views

2answers
283 views

Daunting series of integrals: $\sum_{n=2}^\infty\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log(\frac{1-\sin x}{1+\sin x})dx$

My coleague showed me the following integral yesterday I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
3answers
543 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 ...
2answers
352 views

What does shear mean?

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both ...