# An Introduction to Topology & Homotopy by Allan J. Sieradski

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Example text

Back-substituting = t − t into the first expression and rearranging terms, we obtain that exp(−3µ2 t) sin(µx + κ) = exp(−3µ2 t) sin(µx + κ), in other words, I is an invariant. To verify the group action property for the variable x, set x1 = x( ). Note that sin(µx1 + κ) = exp(3µ2 ) sin(µx + κ) and sin(µx1 (δ) + κ) = exp(3µ2 δ) sin(µx1 + κ) and thus sin(µx1 (δ) + κ) = exp(3µ2 δ) sin(µx1 + κ) = exp(3µ2 δ) exp(3µ2 ) sin(µx + κ) = exp(3µ2 ( + δ)) sin(µx + κ) = sin(µx( + δ) + κ) so that x1 (δ) = x( + δ) as required (for small enough δ and ).

A second and simpler line of argument is strictly for matrix presentations, while a third treats tangent vectors as linear, first order differential operators. We will need all three. The major theorem we prove is that the set of tangent vectors at any given point g ∈ G is in one-to-one correspondence with the set of one parameter subgroups of G. After a discussion of the exponential map in its various guises, we end the chapter with a discussion of concepts analogous to tangent vectors, one parameter subgroups and the exponential map for transformation groups.

50) φ,j − ux ξj , dx dx where we have denoted by [x] the particular index of differentiation on u whose infinitesimal we are considering. 50) are total derivatives. This is important since typically ξ and φ depend on the dependent variables. 17 Adapt the calculation above to show that in the case u = u(x), and K = [x . . x], with |K| terms, Kx = [xx . . x], with |K| + 1 terms, φKx,j = d d φK,j − uKx ξj . 18 Extend the calculation of the previous exercise to show that if u = u(x, y), K = [x . .