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Jul
24
comment Function Field of Variety and Scheme
@NicolasBourbaki Yes, I mean $k[V]$, sorry. But it is also probably reasonable to interpret $k[X]$ as the ring of global sections of $X$, which is isomorphic.
Jul
24
comment Function Field of Variety and Scheme
Your definition of residue field at a point in a scheme agrees with mine, all that is going on here is that localization is exact. I thought you were asking about the field of rational functions of the corresponding variety - if not then I am somewhat confused about what your question is. The notation k[X]? It denotes the coordinate ring of a variety X over a field k. Anyway I think I am being a little sloppy with notation, but probably the commutative algebra fact I mentioned answers your question.
Jul
24
comment Line integral over a intersection of a cylinder and a plane.
@Smithy One parametrizes a circle using sine (y coordinate) and cosine (x coordinate). Then, given a point on your ellipse, if the x-y coordinates are fixed the z coordinate must satisfy the relationship $x - z = 1$, so we know that $z = x - 1$.
Jul
24
answered Function Field of Variety and Scheme
Jul
22
comment Line integral over a intersection of a cylinder and a plane.
you can observe that the intersection defines a function on the unit circle in the xy-plane, and the second equation tells you the height of that function. But I think it is a good idea to learn how to sketch what is going on here, since the individual steps for deriving the parametrization may change from problem to problem, but the overall idea will be the same (look at the shadow, parametrize it, and determine the height at time t using the given formula.)
Jul
22
comment Line integral over a intersection of a cylinder and a plane.
A helpful way to draw such things is to hold one variable constant, then think about what happens when that variable changes. For example, when $z = 0$, the first equation $x^2 + y^2 = 1$ describes a circle on the xy-plane. What happens as $z$ changes? For the second, hold $y$ constant (which doesn't change the equation), then sketch the corresponding line on the $zx$-plane. Now, what happens here as $y$ changes? It is indeed an ellipse. You only need to draw a rough picture here to guide your parametrization of the intersection. If you prefer to do this completely algebraically...
Jul
22
answered Line integral over a intersection of a cylinder and a plane.
Jul
19
comment Converting GAP groups into SAGE permutation groups.
Never mind, I resolved it.
Jul
19
comment Converting GAP groups into SAGE permutation groups.
Is it a stack-exchange faux pas to ask here how you got gap.SmallGroup to run? I am having some difficulty with this - I have posted a question here: math.stackexchange.com/questions/1366086/…
Jul
19
accepted Computing the shape operator
Jul
18
accepted Lie bracket is part of the intrinsic “geometry”? But I have seen it defined without a metric…?
Jul
18
comment Lie bracket is part of the intrinsic “geometry”? But I have seen it defined without a metric…?
Oh I see - the Levi-Civita connection is defined so that this identity holds.
Jul
18
accepted Question of well-definedness of the Levi-Civita connection?
Jul
18
asked Lie bracket is part of the intrinsic “geometry”? But I have seen it defined without a metric…?
Jul
17
revised Question of well-definedness of the Levi-Civita connection?
deleted 1 character in body
Jul
17
revised Riemannian Geometry notational tricks or alternatives
added 22 characters in body
Jul
17
asked Question of well-definedness of the Levi-Civita connection?
Jul
17
asked Riemannian Geometry notational tricks or alternatives
Jul
17
comment When $X\times (Y\times Z)=(X\times Y)\times Z$ in product topology?
@CameronWilliams Wait, so now I'm confused. Does this mean that a proof applying the universal property several times won't work?
Jul
17
answered When $X\times (Y\times Z)=(X\times Y)\times Z$ in product topology?