I'm studying differential forms, and in my homework I'm asked to show that the product of two manifolds $M \times N$ is orientable if and only if both $M$ and $N$ are orientable.  I want to show this using volume forms. For the backwards implication ($\Leftarrow$):  Suppose $M$, m-manifold, and $N$, n-manifold, are orientable. Then there exist nowhere vanishing top forms $\omega_1 \in \Omega^m(M)$ and $\omega_2 \in \Omega^n(N)$. Define $\omega_1 \times \omega_2$ in $M \times N$ by $\omega_1 \times \omega_2(X_1,...,X_m,Y_1,...,Y_n)=\omega_1(X_1,...,X_m)\omega_2(Y_1,...,Y_n)$, and one can easily see that this is a form, a top, nowhere vanishing $(m+n)$-form in $M\times N$. It follows that $M\times N$ is orientable.  My problem is in the forward implication. How do I show that if $M\times N$ is orientable, then so are $M$ and $N$, or, equivalently, if $M$ and $N$ are both not orientable, then $M\times N$ can't be orientable? How can I construct volume forms on $M$ and $N$ from a given volume form in $M\times N$?
I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable.   I am able to prove that the product is orientable if components are orientable (chart is $\{(U_\alpha\times V_{\beta},\phi_\alpha\times \psi_\beta):(\alpha,\beta)\in A\times B \}$, and $\det J=\det J_1 \det J_2&gt;0$ by Cauchy-Binet's theorem), but I don't know how to prove the other direction.   So why this holds: if $M\times N$ is orientable, then $M$ and $N$ are orientable?  Thanks in advance.
What are the different ways that a message can be displayed to a bash shell after a user logs in? I have a CentOS 5.7 VPS using bash as its shell that displays a branded greeting immediately after logging in via SSH. I've been trying to modify it, but can't seem to find where it is in the usual places. So far I've looked in the motd file and checked sshd_config for banner file settings. A banner file is not set.  Where else can I look for where the login message might be? What is a "hat"? What exactly is a hat?  I have seen a lot of people arguing the significance of hats and the limited edition hats. I know the hats are sort of linked to the Winterbash stuff, but what exactly is a hat, what does it do, how do I get one?  Is there a full list of hats? UI annoyances in the suggested edit review on Stack Overflow It is possible to press ⎇ Backspace on macOS in order to delete the previous word (left of the cursor), and ⌘ Backspace to delete the previous line (left of the cursor). 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Is there a way to set the material properties for frame 1 and use the same or rest of the frames while the corresponding meshes are imported afresh from that ones I provide? What is the meaning of 'though'? Okay, of course I've used different popular online dictionaries and the translation makes sense in the most cases.    Sometimes I find usages of 'though' where I don't get the meaning of the word.   An example I've found on twitter     That face, though. [Image of a dog looking weird]   The tweet came without any context or conversation.  When I translate this example to my native language - german - this makes absolutely no sense. At least for me.  So is the above usage some kind of slang? What is the meaning of though in this case?  For the completeness an example that makes absolutely sense for me     Though he tried very hard, he failed the course. Does the scientific community consider the Loschmidt paradox resolved? If so what is the resolution? 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Vortex in liquid collects particles in center At xmas, I had a cup of tea with some debris at the bottom from the leaves.  With less than an inch of tea left, I'd shake the cup to get a little vortex going, then stop shaking and watch it spin.  At first, the particles were dispersed fairly evenly throughout the liquid, but as time went on (and the vortex slowed, although I don't know if it's relevant) the particles would collect in the middle, until, by the time the liquid appeared to almost no longer be turning, all the little bits were collected in this nice neat pile in the center.  What's the physical explanation of the accumulation of particles in the middle?  My guess is that it's something to do with a larger radius costing the particles more work through friction... $|f'(x)| \le g(x)$ implies $|f(b) - f(a)| \le \int_a^b g(x) dx$, without assuming $f'$ to be integrable. What is LINQ? I know it's for databases, but what does it do? 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I am not looking for the I/O scheduler, but the process scheduler, and please, do not refer to guides or howtos for what a scheduler is, what I'm simply asking is if there is a command or file I can run/check to see this. I could check the doc for my kernel but again I'm just wondering if there is a command or file i can check.  I might have been a little redundant above but that's because the posts I've seen people seem to confuse process scheduler with the i/o scheduler, and they don't seem to answer the actual question, just provide links to what a scheduler is which is not what I'm after here, so apologies if the above comments sounded rude. What is the historic present tense? I learned that the historical present may be used to create an effect of immediacy in narratives. I have a question about the historic present tense. Why don't we use the progressive tense instead of the simple present tense?   ※My English skill is poor, so my question might sound rude. But I don't mean to offend anyone. I'd be glad if someone could reply to my question. Doubling the tails in two-sample permutation test What do the subscripts in expectations mean in loss function Can someone explain me what this equation means? I read it from page 152 on .