Fellow geometry enthusiasts:  I am studying a particular geometry and I would like to change from one set of coordinates to a new set that makes the metric unimodular (i.e. it's determinant is one). The components of the metric are rather complicated functions of the coordinates, so I don't know how to solve the system of linear first order PDEs that I get from the Jacobian. Is anyone aware of any software that can give me the new coordinates in terms of the old ones given the Jacobian of the transformation? I have Maple 18, but I have not been able to find this capability in Maple.
What is some good free software for doing physics calculations?  I'm mainly interested in symbolic computation (something like Mathematica, but free).
Prove that the power set of an $n$-element set contains $2^n$ elements Theorem. Let $X$ denote an arbitrary set such that $|X|=n$. Then $|\mathcal P(X)|=2^n$.      Proof. The proof is by induction on the numbers of elements of $X$.      For the base case, suppose $|X|=0$. Clearly, $X=\emptyset$. But the empty set is the only subset of itself, so $|\mathcal P(X)|=1=2^0$.      Now the induction step. Suppose $|X|=n$; by the induction hypotesis, we know that $|\mathcal P(X)|=2^n$. Let $Y$ be a set with $n+1$ elements, namely $Y=X\cup\{a\}$. There are two kinds of subsets of $Y$: those that include $a$ and those that don't. The first are exactly the subsets of $X$, and there are $2^n$ of them. The latter are sets of the form $Z\cup\{a\}$, where $Z\in\mathcal P(X)$; since there are $2^n$ possible choices for $Z$, there must be exactly $2^n$ subsets of $Y$ of which $a$ is an element. Therefore $|\mathcal P(Y)|=2^n+2^n=2^{n+1}$. $\square$      From the above explanation, I don't understand why the set that contains $\{a\}$ will contain $2^{|n|}$ elements when it should clearly be $2^{|1|}$.  The construction of a new set $S$ is the union of the old set with cardinality $n$ and an new element $\{a\}$, therefore the set that do not contain $\{a\}$ still has cardinality $n$ and the set that contains $\{a\}$ is just $\{a\}$, one element...  Can someone please elucidate? Is there a way to get DateTime value from timestamp type column? Pivot in sql server Hubble's constant $a(t)$ appears to be changing over time. The fine stucture constant $\alpha$, like many others in QFT, is a running constant that varies, proportional to energy being used to measure it. Therefore, it could be agued that all running constants have 'evolved' over time as the Universe has expanded and cooled. Both the local and global curvature of the Universe changes over time implying that so too does the numerical value of $\pi$. All these things are however constants (well, let's say parameters since they are not really 'constant'.)   In  with astronomer Sir Fred Hoyle, Feynman said "what today, do we not consider to be part of physics, that may ultimately become part of physics?" He then goes on to say "..it's interesting that in many other sciences there's a historical question, like in geology - the question how did the Earth evolve into the present condition? In biology - how did the various species evolve to get to be the way they are? But the one field that hasn't admitted any evolutionary question - is physics."  So have the laws of physics remained form-invariant over the liftetime of the Universe? Does the recent understanding of the aforementioned not-so-constant constants somehow filter into the actual form of the equations being used? 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But if not, it looks strange. Should I leave it, or you have suggestions for this? "Protect Questions" privilege page has confusing wording The wording on the  privilege page is confusing and would benefit from being clarified. See these questions on  and . Suggested wording from  is to replace this:  Which questions can be protected? Any question, even a deleted question, can be protected and unprotected, subject to the following restrictions:  The question must be at least a day old A user can unprotect any protected question   with the following:  Which questions can be protected? Any question at least a day old, even a deleted question, can be protected and unprotected. A user with the privilege can unprotect any protected question. How do question bookmarks work? What does bookmarking a question mean? How do you bookmark a question? Do users get updates on their bookmarked questions? Are my bookmarks publicly visible? How do you see your own bookmarks? 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I just thought about the substitution $x \leftrightarrow\frac{1}{1+x^2}$ and I get the  following result:  $$I(a) = \frac{\cos(1/2)}{2^{2-a}} + \int_0^1\frac{2x^3 \cos\left(\frac{1}{1+x^2}\right)}{(1+x^2)^{3-a}} \, dx - \int_0^1 \frac{2x^3\sin\left(\frac{1}{1+x^2}\right)}{(1+x^2)^{2-2a}}\,dx$$ If I could prove that the last summands are $o(I(a))$ then it would be OK, but I think, in general my strategy won't work. Apparently this is supposed to be an easy exercise, but I can't come up with anything right now. Does using an extender actually change the aperture of the lens? I just bought a Canon 2X extender and I always believed that it would only result in 2 stops' loss of light and not change the aperture.    However, when I tested this extender on a Canon 135mm f/2 lens, the maximum aperture I could set went down to F/4.  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