Greg Graviton
  • Member for 11 years, 4 months
  • Last seen more than a month ago
20 answers
165 votes
27k views
Striking applications of integration by parts
145 votes

I always liked the derivation of Taylor's formula with error term: $$\begin{array}{rl} f(x) &= f(0) + \int_0^x f'(x-t) \,dt\\ &= f(0) + xf'(0) + \int_0^x tf''(x-t)\,dt\\ &...

View answer
6 answers
165 votes
29k views
An Introduction to Tensors
58 votes

Mathematicians and physicists use very different languages when they talk about tensors. Fortunately, they are talking about the same thing, but unfortunately, this is not obvious at all. Let me ...

View answer
9 answers
90 votes
29k views
Learning mathematics as if an absolute beginner?
46 votes

This is a difficult question to answer, mainly because any advice must be very personal to be useful for you. I'll try anyway. Before learning mathematics, one has to learn how to learn mathematics. ...

View answer
7 answers
77 votes
23k views
Quotient ring of Gaussian integers
34 votes

Go back one step and add the defining equation for $i$ to the ideal. In other words, consider your ring as a quotient of the ring of polynomials $\mathbb Z[x]$: $$ \mathbb Z[i] / (3-i) = \mathbb Z [x]...

View answer
3 answers
21 votes
11k views
How do you find the Lie algebra of a Lie group (in practice)?
Accepted answer
28 votes

Here a worked out example: What is the Lie algebra of the group of rotations in 3-dimensional space, $SO(3)$? Matrices $A\in SO(3)$ are defined by the property that they are invertible and that the ...

View answer
2 answers
21 votes
3k views
Farkas’ lemma: purely algebraic intuition
25 votes

If you want an intuition that explains why Farka's Lemma should be true, you will have to use the geometric interpretation; there's no way around that. If you want an intuition that shows what Farka'...

View answer
3 answers
48 votes
10k views
Simple explanation of a monad
15 votes

Monads in Haskell and monads in category theory are very much the same: A monad consists of a functor $T: C \to C$ and two natural transformations $\eta_X : X \to T(X)$ (return in Haskell) and $\mu_X :...

View answer
3 answers
14 votes
4k views
Do trivial homology groups imply contractibility of a compact polyhedron?
Accepted answer
12 votes

To sum up the comments: when Poincaré worked on the beginnings of algebraic topology, he originally thought that a space with trivial homology groups must be contractible. (More precisely, he thought ...

View answer
3 answers
10 votes
4k views
Definition of the gradient for non-Cartesian coordinates
Accepted answer
11 votes

It turns out that there are two different but related notions of differentiation for a function $f:\mathbb R^n\to\mathbb R$: the total derivative $df$ and the gradient $\nabla f$. The total ...

View answer
11 answers
37 votes
15k views
What is the (mathematical) point of straightedge and compass constructions?
11 votes

The beauty of straightedge and compass constructions, as opposed to the use of, say, a protractor, is that you don't measure anything. With ruler and compass you can bisect an angle without knowing ...

View answer
5 answers
58 votes
7k views
Is there a definition of determinants that does not rely on how they are calculated?
11 votes

Determinants are best understood in the context of exterior algebra, which goes back to the work of Hermann Grassmann. Here a down-to-earth description of the intuition behind it. Consider an $n$-...

View answer
1 answers
5 votes
3k views
Proving the Shoelace Method at the Precalculus Level
Accepted answer
9 votes

One way is to note that $x_1y_2 - x_2y_1$ is a signed area, i.e. it may positive or negative. Adding up all the signed areas of the triangles formed by the points $O$, $P_k$ and $P_{k+1}$ will cancel ...

View answer
2 answers
17 votes
604 views
Constructing a number not in $\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\epsilon}{2^k},q_k+\frac{\epsilon}{2^k})$
8 votes

I'll answer the first question by considering a slightly different problem where it is easier to explicitly construct an irrational number not in $S_\epsilon$. Namely, consider the interval $(0,1)$. ...

View answer
3 answers
13 votes
1k views
How to show determinant of a specific matrix is nonnegative
Accepted answer
8 votes

It was pointed out in the comments that you can interpret the matrix as a bilinear form and obtain the inequality $$ x^T A x = \cos(\pi/N) \sum_{i=1}^N x_i^2 - \sum_{i=1}^N x_i x_{i+1} \cos\theta_i \...

View answer
2 answers
3 votes
347 views
Preserving the extrema of one function after applying another
Accepted answer
6 votes

You don't need to assume that $g$ is non-zero, and it could be strictly decreasing as well. Furthermore, the conditions on $g$ only need to hold on the image of $f$ (which doesn't need to the be whole ...

View answer
1 answers
6 votes
208 views
Estimating $\#\{\{\alpha k\} < 1/\sqrt{k} : k \leq n\}$ for irrational $\alpha$
Accepted answer
5 votes

The Weyl equidistribution theorem says that for irrational $\alpha$ and sufficiently many $k$, the fractional parts $\{\alpha k\}$ will be equidistributed in the interval $[0,1]$. To apply this to ...

View answer
1 answers
1 votes
994 views
How can I characterize the type of solution vector that comes out of a matrix?
Accepted answer
5 votes

If all entries of $A^{-1}$ are positive numbers, then $A$ has the property you desire. (Edit: This condition is both sufficient and necessary. If one entry $A^{-1}_{ij}$ is negative, then the choice $...

View answer
1 answers
6 votes
483 views
Validating a mathematical model (Lagrange formulation and geometry)
Accepted answer
4 votes

Concerning the physical meaning, I take it that $f_1$ and $f_2$ represent the fractions of the two phases in the alloy (this implies $f_1 + f_2 = 1$). I imagine $x_1$ and $x_2$ to correspond to an ...

View answer
2 answers
5 votes
2k views
Does there exist a compact Hausdorff topology on the natural numbers?
4 votes

There exists a bijection between the one-point compactification $\mathbb{N}\cup\lbrace\infty\rbrace$ and $\mathbb N$, for instance by mapping $$ \infty \mapsto 0 \text{ and } n \mapsto n+1 .$$ Use ...

View answer
2 answers
4 votes
1k views
Divergence Theorem, Laplacian, Energy Minimization
Accepted answer
4 votes

The identity $$ \int_V d\vec x\, \langle \nabla \phi, \nabla u \rangle = -\int_V d\vec x\ \phi\Delta u $$ is a consequence of integration by parts and the divergence theorem. Namely, we have $$ \...

View answer
2 answers
7 votes
2k views
How to draw a complex line bundle?
Accepted answer
4 votes

Apparently, Mario Serna has produced pictures of $U(1)$-bundles on his webpage and in his paper "Riemannian Gauge Theory and Charge Quantization". Here an example The image represents a trivial $\...

View answer
3 answers
3 votes
7k views
Why we need frequency domain?
Accepted answer
3 votes

I don't know much about its utility in economics, but Fourier analysis is invaluable in physics. The main reason for that is that it transforms linear differential equations into simple algebraic ...

View answer
4 answers
12 votes
2k views
Are translations of a polynomial linearly independent?
3 votes

Here a fancy proof that uses operators and differentiation. Let $T_a$ be the translation operator which maps polynomials to their translates $$ T_a p(x) = p(x + a) .$$ Restricting attention to ...

View answer
6 answers
16 votes
3k views
Historical textbook on group theory/algebra
3 votes

Algebra is a very large field, so you probably want to be a bit more specific. In case you are wondering about Galois Theory, and want to learn its history for the purpose of understanding it, I ...

View answer
4 answers
7 votes
1k views
Indefinite summation of polynomials
3 votes

As other answers have noted, you are about to discover the calculus of finite differences. For practical calculations, here a most useful fact: the rule $\frac{d}{dx} x^n = n x^{n-1}$ corresponds ...

View answer
2 answers
3 votes
540 views
Implicit Differentiation: Multiple Solutions?
Accepted answer
2 votes

Since it's an equation of degree 2, the curve in question is a conic section. Looking at the coefficients, it's probably an ellipse. (The criterion for that is that the quadratic form should always be ...

View answer
2 answers
7 votes
8k views
Fundamental group of a torus with points removed
Accepted answer
2 votes

Hint: removing $n$ points will also give something that only consists of 1-dimensional things. You can also use van Kampen's theorem to calculate the fundamental group directly. EDIT: Err, you can't, ...

View answer
3 answers
8 votes
472 views
Finite groups: $H \leq A \times B$. Is $H \cong C \times D$ for some $C \leq A$, $D \leq B$?
2 votes

It's not possible if you want the isomorphism to be compatible with the product structure. Namely, choose $A=B$ and $H = \lbrace (a,a) : a\in A\rbrace$ the "diagonal subgroup". Clearly, the subgroup $...

View answer
1 answers
1 votes
147 views
An inequality about norms of vector fields on Riemannian manifolds
2 votes

This is simply the triangle inequality: $$ \lVert\frac{dC}{dt}\rVert = \lVert\frac{dC}{dt} + \beta - \beta\rVert \le \lVert\frac{dC}{dt} + \beta \rVert + \lVert-\beta\rVert < N + \lVert\beta\rVert ...

View answer
5 answers
13 votes
2k views
Visualising $\mathbb CP^2$: a problem of attaching cells with a dimension gap >1
2 votes

Concerning the Hopf fibration, the film "Dimensions" does a superb job of visualizing it (chapters 7-8 in the table of contents). Concerning the visualization of the glueing, maybe it helps to ...

View answer