# Tag Info

### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 "...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...

### 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 '...
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### 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 ...

### 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{...
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### 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 ...
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### 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 ...
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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\}=\...