By D. Martin (Auth.)
This account of uncomplicated manifold idea and worldwide research, in response to senior undergraduate and post-graduate classes at Glasgow collage for college students and researchers in theoretical physics, has been confirmed over decades. The therapy is rigorous but much less condensed than in books written essentially for natural mathematicians. must haves contain wisdom of uncomplicated linear algebra and topology. Topology is integrated in appendices simply because many classes on arithmetic for physics scholars don't comprise this subject.
- Provides a entire account of easy manifold conception for post-graduate students
- Introduces the fundamental idea of differential geometry to scholars in theoretical physics and mathematics
- Contains greater than one hundred thirty workouts, with important tricks and solutions
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Extra resources for Manifold Theory. An Introduction for Mathematical Physicists
Example text
Be an element of an orthonormal basis of A''(V„). Then *ω is defined by ( /7-form e Λ... 1 1 (e'
E. basis-free) definition of contraction can be given in the following way. e. ®a> , where u eV„ ( i = l , . . , r ) , (o'eV* = Then the tensor ω ( « ) « 2 ® · · ·®« ®ω ®.. ®ω\ for example, is of type ( r - 1, s — 1) and is a contraction of T. ω * , 1 • u i 2 . . I, "h-l, Λ ' in agreement with the definition of contraction in terms of components. Since any tensor of given type is a linear combination of decomposable tensors of the same type, for example, the appropriate basis elements, the definition of contraction just given for a decomposable tensor can be extended to an arbitrary tensor.
We now define four operations on tensors, namely, addition, multiplication b y a y s c a l a r , o u t e r m u l t i p l i c a t i o n and c o n t r a c t i o n (which includes i n n e r m u l t i p l i c a t i o n ) . x V„ , x to IR. r factors s factors Let S and Τ be tensors of type (r,j) and let (o'eV* ( i = l r), M,eV„ (i = 1 s). Then, by definition. S + Τ is the tensor of type (r,s) for which (S + Τ) (ω 1 Since ω', Μ , w ) = S(a) ω', u, u) + Τ(ω' ω', «ι Μ ) . 7: = S(e'-' e\ e e,) h and similarly for T, we immediately obtain the rule for the addition of tensors (of the same type) in terms of components, namely, (5 + T)'^ ;'•· = 5^' + TJl; 'f .