By Dr. Walter Thirring, Dr. Evans Harrell (auth.)

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**Example text**

Henceforth, the IVi II gjj 11/2 will be called the componentst of v, because of their intuitive significance. When written out in these components, many formulas lose their simplicity. A diffeomorphism

Ir are the components of the tensors, and are the quantities usually called tensors in physics. The tensor product of r arbitrary vectors is defined t Following the usual convention we use subscripts for the bases and superscripts for the components in a tangent space, and do it the other way around in a cotangent space. This does not fix what to do about coordinates, which are not vectors. 4 Tensors similarly to the tensor product of basis elements. Not every tensor can be written in the form VI ® V2 ® ...

18; 2). , $* X = T($) X $-1. 0 t If these fields are Xi' 0 i = 1, ... , m, this means that the Xi(q) are lin(;llriy independent Vq E M. 22) 1. M = IRn, <1>: x -+ x + a, T(**X: x -+ (x, vex - a». A vector remains unchanged under a displacement, but one must take care to talk only about a vector at one particular point, which has different coordinates in the new system. 2. M = IRn, :Xj -+ LjkXk, T(*X:Xj-+ (Xi' LikVk(L -lX». *