Recent content by madshiver

My professor ended up defining the tangent space as an affine vector subspace, which actually kind of matches the geometric intuition and in a sense allows one to calculate the length, direction and "foot" of the vector. Then again, we did not really prove anything with the tangent space, just...

Ok, this makes sense. I guess the description by Lavinia corresponds to the construction mentioned by jgens. Thank you again ;).

Thank you for your reply. I realize that on every point it is a different function. The condition \omega (p) \in \bigwedge^k(\mathbb R^n) would express this fact, except that the new function acts on the actual vector space and not on the tangent space. But if my analogy to that of vector...

I have been working through Spivak's fine book, but the part about differential forms and tangent spaces has left me confused. In particular, Spivak defines the Tangent Space \mathbb R^n_p of \mathbb R^n at the point p as the set of tuples (p,x),x\in\mathbb R^n. Afterwards, Vector fields are...

Thanks :).

I have a question regarding phase transitions: It all textbook I've seen the assumption that dP=0 during a phase transition (so that for example dH=TdS is true during a phase transition). Could anyone explain why this should be true? I guess I could relate this question to a question I...

I'm in the second year of my studies of molecular biology and in the first year of my mathematics degree (in most german universities the concept of a "double major" does not really exist, so you need to "enroll" a second time in order to be granted two degrees at once). I'd certainly like to...

Having grown up in Greece and then moving to Germany left me utterly confused. I used to write the greek alpha symbol the same way I wrote "a". And then suddenly they were writing a fish-like symbol and that was supposed to be an alpha and the same exercise had both "alpha" and "a"... This took...

Despite those insightful replies, I am still not really sure where the line should be drawn. So, although it is clear that research generally can't make up for bad grades, how much does it help? For example would it be worth it to sacrifice a 3.9-4.0 GPA for a 3.7 one in order to also get some...

In order to give a quick background: I am a first-year university student. So far pretty much all I have been doing is to study from textbooks, solve problems, etc. Thus I have been able to attain A's in all exams so far. Yet, the question pops up whether it would be more beneficial (beyond the...