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1500 questions
67
votes
5 answers
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
I have a question:
Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that
$$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$
How can I do this? Don't know how to start.
James
- 689
67
votes
1 answer
Effect of elementary row operations on determinant?
1) Switching two rows or columns causes the determinant to switch sign
2) Adding a multiple of one row to another causes the determinant to remain the same
3) Multiplying a row as a constant results in the determinant scaling by that constant.
Using…
dfg
- 4,071
67
votes
4 answers
How to calculate the pullback of a $k$-form explicitly
I'm having trouble doing actual computations of the pullback of a $k$-form. I know that a given differentiable map $\alpha: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ induces a map $\alpha^{*}: \Omega^{k}(\mathbb{R}^{n}) \rightarrow…
Tony Burbano
- 671
67
votes
4 answers
How to tell if a set of vectors spans a space?
I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, c_3, c_4$ such that $(x, y, z) = c_1(1, 1, 1) +…
Javier
- 7,444
67
votes
8 answers
Is there a way to get trig functions without a calculator?
In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a calculator?
Sometimes I don't feel right when I can't do…
Jonathan Lam
- 853
67
votes
10 answers
Is it not effective to learn math top-down?
By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving basic results. Eventually you'll get there but is…
Daniel Donnelly
- 22,288
67
votes
7 answers
If eigenvalues are positive, is the matrix positive definite?
If the matrix is positive definite, then all its eigenvalues are strictly positive.
Is the converse also true?
That is, if the eigenvalues are strictly positive, then matrix is positive definite?
Can you give example of $2 \times 2$ matrix with…
user957
- 3,457
67
votes
2 answers
Why is $\varphi$ called "the most irrational number"?
I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio of integers. What is meant by most irrational?…
Christopher King
- 10,773
67
votes
7 answers
Is there any geometric intuition for the factorials in Taylor expansions?
Given a smooth real function $f$, we can approximate it as a sum of polynomials as
$$f(x+h)=f(x)+h f'(x) + \frac{h^2}{2!} f''(x)+ \dotsb = \sum_{k=0}^n \frac{h^k}{k!} f^{(k)}(x) + h^n R_n(h),$$
where $\lim_{h\to0} R_n(h)=0$.
There are multiple ways…
glS
- 7,963
67
votes
1 answer
Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$
Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$
Or equivalently:
$$\boxed{\int_0^\pi\frac{x\ln(1-\sin x)}{\sin…
Zacky
- 30,116
67
votes
1 answer
Abstract nonsense proof of snake lemma
During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal properties. It was an interest little shared by…
Jesko Hüttenhain
- 15,116
67
votes
3 answers
Why is the localization at a prime ideal a local ring?
I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is the localization of the ring $ A $ with respect to…
Bryan
- 887
67
votes
8 answers
How much Math do you REALLY do in your job?
I am writing this, as I am a currently an intern at an aircraft manufactur. I am studying a mixture of engineering and applied math. During the semester I focussed on numerical courses and my applied field is CFD. Even though every mathematician…
Thomas
- 4,441
67
votes
5 answers
Is there any branch of Mathematics which has no applications in any other field or in real world?
Is there any branch of Mathematics which has no applications in any other field or in real world ?
for instance , maybe : number theory ? mathematical logic ?
is there something like this ?
FNH
- 9,440
67
votes
4 answers
Counterexample Math Books
I have been able to find several counterexample books in some math areas. For example:
$\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted
$\bullet$ Counterexamples in Topology, Lynn Arthur Steen, J. Arthur Seebach…
Amzoti
- 56,629