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August 15

Projective Space

When you apply the construction of a projective space to Rn, by removing the origin and identifying all points that are constant multiples of each other as vectors, the space you get resembles R(n-1) plus a bit. This "bit", in the case n=2, seems to be beyond the other numbers of R1 in a way that resembles infinity, so it's convenient to call it the point "at infinity". This terminology behaves well with respect to the field operations. So, that, for instance, infinity times anything besides zero is itself. That extends into more dimensions, and into other fields, like the complex numbers. The question I have is, how well does it extend to fields with positive characteristic, especially the finite fields? In this case, there is no real sense of "size" to anything, so is there any sense in which the new element or subspace is "at infinity"? Black Carrot (talk) 05:29, 15 August 2008 (UTC)[reply]

Your point about the field operations still holds, I believe. Algebraist 09:31, 15 August 2008 (UTC)[reply]

Yes, you can do the same construction as well. Although for finite fields the topological aspect vanishes and the algebraic one seems rather poor, still the combinatoric is very rich. Geometrically, you can start from a vector space V over a field K: the corresponding projective space (that you can identify with the set of 1 dimensional subspaces of V) is of great interest also in case of finite fields, e.g. in combinatorics. Notice that all lines in V are similar to each other (in a sense that one could make precise), thus no element of the projective space is "at infinity" more than others. On the other hand, if you also fix a hyperplane W of V (say, an affine hyperplane not containing the origin) then you can see the projective space as a superset of W (identifying any point of W with a line of V in the obvious way), and you have the right to say that what is out of W is "at infinity". But it seems to me that the corresponding enlargement of a finite field, e.g. Z_2, by means of a point at infinity is somehow of less algebraic interest. 79.38.22.37 (talk) 14:47, 15 August 2008 (UTC)[reply]

Well, I know they're of some theoretical interest in algebra, since some of the finite simple groups are represented as projective matrices over finite fields. I'm especially curious about infinite fields with finite characteristic, though, which can be given interesting topologies. Does anyone know of an interpretation of one of these topologies, that would be consistent with the interpretation of points in its projective space being at infinity? Perhaps a topology based on a metric, like in the complex numbers, where division by a deleted neighborhood of zero gives a deleted neighborhood of infinity. Black Carrot (talk) 12:08, 16 August 2008 (UTC)[reply]

First Principle

How to find the derivative of $x^{3}\cos x$ USING FIRST PRINCIPLE? —Preceding unsigned comment added by Ftbhrygvn (talk • contribs) 09:38, 15 August 2008 (UTC)[reply]

Write down the definition of the derivative and work it out. You'll want the addition formula for cos and the fact that sin(x) is approximately x for small x, which can both be proven geometrically. Algebraist 10:18, 15 August 2008 (UTC)[reply]

Just to be more clear, in general when you see using first principles, means the problem can't be solved using all of the nice tricks you've learned for doing derivatives (like the chain rule) and are forced to use the original limit formula for a derivative. As the above poster hinted at, in order to do the limit, you'll need some trigonometric identities. Anythingapplied (talk) 13:22, 15 August 2008 (UTC)[reply]

...although in this case the problem can be easily solved using the product rule and a knowledge of the derivatives of trigonometric functions, so I imagine the "first principles" instruction is just to give the reader experience of taking derivatives "by hand", so that they will appreciate the usefulness of the "tricks". Gandalf61 (talk) 13:32, 15 August 2008 (UTC)[reply]