By Gerardo F. Torres del Castillo
ISBN-10: 0817682708
ISBN-13: 9780817682705
ISBN-10: 0817682716
ISBN-13: 9780817682712
This textbook explores the idea in the back of differentiable manifolds and investigates quite a few physics functions alongside the way in which. simple recommendations, equivalent to differentiable manifolds, differentiable mappings, tangent vectors, vector fields, and differential types, are in brief brought within the first 3 chapters. bankruptcy four offers a concise creation to differential geometry wanted in next chapters. Chapters five and six offer fascinating purposes to connections and Riemannian manifolds. Lie teams and Hamiltonian mechanics are heavily tested within the final chapters. integrated in the course of the ebook are a suite of routines of various levels of trouble.
Differentiable Manifolds is meant for graduate scholars and researchers drawn to a theoretical physics method of the topic. must haves comprise multivariable calculus, linear algebra, differential equations, and a simple wisdom of analytical mechanics.
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Extra info for Differentiable Manifolds: A Theoretical Physics Approach
Sample text
Thus, from the foregoing relations we find that y= 4c , 4 + c2 (t − t0 )2 x=− 2c2 (t − t0 ) . 1 One-Parameter Groups of Transformations and Flows = 35 (x0 − (x0 2 + y0 2 )t/2, y0 ) . 12) corresponds to a one-parameter group of transformations. 41. 10 Find the integral curves of the vector field X = 1 ∂ (x ∂x x 2 +y 2 ∂ − y ∂y ) on R2 \ {(0, 0)} and the one-parameter group of diffeomorphisms generated by X. 10) one notices that if one looks for the integral curves of f X, where f is some real-valued differentiable function, on eliminating the variable t the function f disappears and one obtains the same equation for dy/dx as obtained in the preceding example.
Yk . 45). 30 Show that if X, Y ∈ X(M) and t ∈ Tk0 (M), then £X (£Y t) − £Y (£X t) = £[X,Y] t. 31 Show that if X ∈ X(M) and t ∈ Tk0 (M), then £X (X t) = X (£X t). 32 Let t be a differentiable tensor field of type ( kl ) on M. Assuming that the first k arguments of t are covectors and defining £X t by (£Xt)(α1 , . . , αk , Y1 , . . , Yl ) ≡ X t (α1 , . . , αk , Y1 , . . , Yl ) k − t (α1 , . . , £Xαi , . . , αk , Y1 , . . , Yl ) i=1 l − t (α1 , . . , αk , Y1 , . . , £XYi , . . , Yl ), i=1 for α1 , .
In some cases ϕ is not defined for all t ∈ R, and for that reason it is not a oneparameter group of transformations. However, for each x ∈ M there exist a neighborhood, U of x and an ε > 0 such that ϕ is defined on U × (−ε, ε) and is differentiable. The map ϕ is called a flow or local one-parameter group of transformations and X is its infinitesimal generator. If X is the infinitesimal generator of a one-parameter group of transformations or a flow, the transformations ϕt are also denoted by exp tX.
Differentiable Manifolds: A Theoretical Physics Approach by Gerardo F. Torres del Castillo
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