Tensors

gentillabdulla

I have a few questions on tensors.

How do you do basic operations on them?

My intuition tells me you could write them by their components that and treat them like a matrix and do the said operations.(Someone would need to check this.)

Such as the Riemann tensor.

We have a Riemann tensor Riemann(u,v,w). If we write it as a matrix then we can have the components of each individual vector and then multiply it by another tensor, also written as a matrix, Riemann(r,g,c). What I am trying to say is that can it be written as a matrix and then been operated on?

ANY help would be appreciated.

equivariant

http://en.wikipedia.org/wiki/Tensor#Tensor_product

This explains how tensor products work in terms of coordinates. Note that matrix multiplication is not an example of tensor product. Rather it is an example of tensor product followed by a contraction.

sponsoredwalk

Hey gentillabdulla you might appreciate working through this book first:

Schaum's Outline of Vector Analysis with an Introduction to Tensor Analysis.

If you use the "look inside" amazon feature you'll see that the book
applies multivariable calculus/linear algebra to things like mechanics and
even differential geometry which would be helpful in reading things on
General Relativity. I doubt this book is of the utmost rigour but it can't
hurt to get a fairly intuitive introduction to tensors knowing you have the
prerequisites. I don't know how much you've studied but this book looks
as though it requires no more than single variable calculus and some
linear algebra.

If this is a problem for you, or you find learning from a schaums book
difficult, which I'll bet is possible, you should think of getting Serge Lang's
Calculus of Several Variables which would be more than enough to be able
to conquer this book.

Just a suggestion

Anonymo

gentillabdulla, it is possible to expand a tensor in terms of it's components. since in gr the general assumption is that you've a manifold this means that locally you can choose a basis. when you write a tensor out in terms of it's components you are essentially writing it out in terms of a basis. That's all fine but really what you're doing there is writing a globally defined quantity in terms of local quantities. A better language to understand tensors may be in terms of multilinear maps. Then manipulation of tensors is seen to follow the rules of linear transformations on it's components. With a bit of differential geometry you'll learn about things like inner products (though really you've seen this before by contracting with the metric), exterior products and hodge products - which are all better understood without writing them in terms of local coordinates.

i hope this helps, even though it's more of a 'what not to do' rather than a 'what to do'