118 votes

How would you discover Stokes's theorem?

Here's an intuitive way to discover Stokes's theorem. Imagine chopping up the surface $S$ into tiny pieces such that each tiny piece is a parallelogram (or at least, each tiny piece is approximately a ...
littleO's user avatar
  • 52k
111 votes

Expressing the determinant of a sum of two matrices?

When $n=2$, and suppose $A$ has inverse, you can easily show that $\det(A+B)=\det A+\det B+\det A\,\cdot \mathrm{Tr}(A^{-1}B)$. Let me give a general method to find the determinant of the sum of two ...
Drake Marquis's user avatar
104 votes

What is a differential form?

To talk about differential forms, first we need to talk about manifolds and vector fields. Informally speaking, a manifold is any space which is locally Euclidean. That is, the area around every point ...
Plutoro's user avatar
  • 22.7k
94 votes

Notation for partial derivative of functions of functions

This is a long answer, and I do eventually answer your question directly, but first, there are several preliminary matters which need to be addressed. The biggest obstacle you need to overcome is ...
peek-a-boo's user avatar
  • 55.9k
87 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

With respect to the first part of your question: No, a function with two global minima does not necessarily have an additional critical point. A counterexample is $$ f(x, y) = (x^2-1)^2 + (e^y - x^2)^...
Martin R's user avatar
  • 114k
63 votes

Why are proofs not written as collections of logic symbols but are instead written in sentences?

You have not translated the pages from Apostol's book into mathematical logic. What you have done is to transcribe them into your own idiosyncratic shorthand, which may be useful to you but is less ...
David K's user avatar
  • 98.5k
57 votes

What does a triple integral represent?

You can think of the integrand as the "density" of the region and the value of the integral as the "mass" of the object. For example, $$ \int_0^1\int_0^1\int_0^1 1 \, \text{d}x \, \text{d}y \,\text{...
BigbearZzz's user avatar
  • 15.1k
55 votes

Why does a distance and its square reach their minimum at the same point?

If you want to figure out which of $\sqrt{2}$ and $\sqrt{3}$ is the smallest, you can do this simply by comparing $2$ and $3$ instead. More generally, if you want to make a square root as small as ...
Hans Lundmark's user avatar
55 votes

Why not learn the multi-variate chain rule in Calculus I?

I used to think this, too, until I taught Calculus I. If you, as a math student and enthusiast, like to see the product rule, etc., as special cases of the multivariate chain rule, then that is good ...
Matthew Leingang's user avatar
41 votes

Are there other kinds of bump functions than $e^\frac1{x^2-1}$?

There is even simpler example from this book of Loring Tu. You simply start with $$f(t)=\left\{\begin{array}{lr} 0&t\leqslant 0\\ e^{-1/t}&t>0 \end{array}\right..$$ Then you define $$g(t)=\...
Fallen Apart's user avatar
  • 3,785
41 votes

Calculator similar to Desmos but for $3$D

Updated, December 2018: I made the following website with the aim of producing a Desmos-like experience in 3D for my multivariable calculus students. math3d.org You can create and animate points, ...
Chris Chudzicki's user avatar
41 votes

What does a triple integral represent?

When $1$ is your integrand, you have these geometric interpretations: $\quad\int dx$ is to length $\quad\iint dA$ is to area $\quad\iiint dV$ is to volume When your integrand is some function, then ...
zahbaz's user avatar
  • 10.5k
40 votes

Why/How does the determinant of the Hessian matrix, combined with the 2nd derivatives, tell us max., min., saddle points? Reasoning behind it?

Given a smooth function $f: \mathbb{R}^n \to \mathbb{R}$, we can write a second order Taylor expansion in the form: $$f(x + \Delta x) = f(x) + (\nabla f(x))^\top \Delta x + \frac{1}{2}( \Delta x)^\top ...
ಠ_ಠ's user avatar
  • 10.7k
40 votes

What does it mean to multiply differentials?

$\DeclareMathOperator{\Area}{Area}$Edit: In the original answer, I was a bit careless with signed versus unsigned area. The original question implicitly asks about signed area (i.e., area where "...
Andrew D. Hwang's user avatar
38 votes

If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$

We define $f(x,y)=x^{4y^2}+y^{4x^2}$. This is my plan to solve the problem: Since $x+y=1$, we replace $y$ by $1-x$. We make a new function: $g(x)=x^{4(1-x)^2}+(1-x)^{4x^2}$ Therefore, we must find ...
Robin Aldabanx's user avatar
37 votes

Partial derivative in gradient descent for two variables

Despite the popularity of the top answer, it has some major errors. The most fundamental problem is that $g(f^{(i)}(\theta_0, \theta_1))$ isn't even defined, much less equal to the original function. ...
Christian Sykes's user avatar
37 votes

WolframAlpha says limit exists when it doesn't?

This limit is an excellent example to illustrate the power of the (two-)path test and apparently also an excellent example to see that you have to be very careful with how mathematical software deals ...
StackTD's user avatar
  • 27.9k
35 votes

Finding the Center of Mass of a disk when a part of it is cut out.

The center of mass can probably be computed by using difficult integrals, but it can also be computed in a very simple way. Let's call $\vec{OG}$ the center of mass, $m_{big}$ and $m_{small}$ the mass ...
Zakhurf's user avatar
  • 878
33 votes

How "messy" can a multivariable function be?

The granddaddy of ill-behaved functions is $e^{1/x}$. Even in one dimension it is very strange (having what is called an "essential singularity" at the origin), but in 2D or in the complex ...
Yly's user avatar
  • 15.3k
33 votes

The connection between the Jacobian, Hessian and the gradient?

You did not do anything wrong in your calculation. If you directly compute the Jacobian of the gradient of $f$ with the conventions you used, you will end up with the transpose of the Hessian. This is ...
Scott Staniewicz's user avatar
32 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

$ \def\norm#1{\lVert#1\rVert} $The answer to the question as stated is no as Martin showed, but is yes if we add the condition that $f(x)→∞$ as $\norm{x}→∞$. Martin's example pushes the saddle point '...
user21820's user avatar
  • 57.9k
30 votes

Why does a distance and its square reach their minimum at the same point?

This has nothing to do with derivatives, nor epsilons; it's pure logic. If you have a function $f:\>P\to{\mathbb R}_{\geq0}$ defined on some set $P$ (like a parabola in the plane) and a strictly ...
Christian Blatter's user avatar
30 votes

How to derive the divergence theorem from the General Stokes theorem?

Let $\Omega$ be an open subset of $\mathbb{R}^n$ with $\partial\Omega$ of class $\mathscr{C}^\infty$, and let $X$ be a smooth vector field on $\Omega$. Now we compute \begin{align} d(i_X\operatorname{...
Yousuf Soliman's user avatar
30 votes

Why does a saddle remain a saddle when you change coordinates?

Your question is at least partly rather philosophical. I think you're thinking too much about the algebra and too little about the geometry. The mathematical object you are trying to reason about is ...
Ethan Bolker's user avatar
  • 95.5k
29 votes

Is writing the divergence as a "dot product" a deception?

$ \newcommand\PD[2]{\frac{\partial#1}{\partial#2}} \newcommand\tPD[2]{\partial#1/\partial#2} \newcommand\dd{\mathrm d} \newcommand\R{\mathbb R} \newcommand\diff\underline \newcommand\adj\overline \...
Nicholas Todoroff's user avatar
28 votes

What is the solution to Nash's problem presented in "A Beautiful Mind"?

The problem is to find a subset $X$ of $\mathbb{R}^3$ such that if $V$ is the vector space of vector fields $F$ on $\mathbb{R}^3$\ $X$ with $\nabla\times F = 0$ and $W$ is the vector space of vector ...
Enrico M.'s user avatar
  • 26.1k
28 votes

What does shear mean?

Consider a little block in the fluid with edge lengths $a,b,c$. Velocity at the origin is $\vec{v}=(u,v,w)$. With help of a Taylor series expansion we can approximately write the velocity at $\vec{r} =...
Han de Bruijn's user avatar
28 votes

Rigorous proof that $dx dy=r\ dr\ d\theta$

In the geometric approach, $dr^2=0$ as it is not only small but also symmetric (see here). In the algebraic, more rigorous approach, you are deriving $x$ by $\theta$ and $y$ by $r$, but you are ...
geodude's user avatar
  • 8,067
28 votes

Preimage of a compact set

Not generally. The pre-image is closed because $f$ is continuous. But it need not be bounded. The simplest example is to let $f(x,y)=0$ for all $x, y.$ Then $\{0\}$ is compact and $f^{-1}\{0\}=\...
DanielWainfleet's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible