Can someone give me a dummies definition of what a tensor basically is and what its applications are? Thanks
Equation 1 contains the components is the metric tensor. The components in that particular case were all zero except for g11 and g2 which equal 1.mathwonk said:For instance on the first page of the site you just referred us to, equation (1) displays the "metric tensor" in exactly the form I gave for a second order (or rank) tensor, namely it has an expression as a homogeneous polynomial of degree 2.
I think you're confusing the tensor with the expression the components of the tensor appears in. A general tensor is a geometric object which is linear function of its variables which maps into scalars. For example: Let g be the metric tensor. Its a function of two vectors. The boldface notation represents the tensor itself and not components in a particular coordinate system. An example of this would be the magnitude of a vector.Perhaps the confusion is that I was referring to the appearance of a (symmetric) tensor in a given coordinate system, and your sources emphasize the way these representations change, under change of coordinates.
There are two ways of looking at tensors. I've been meaning to make a new web page to emphasize the geometric meaning but am unable to do so at this time. Plus I'm still thinking of the best way to do that.Unfortunately many sources emphasize the appearance or representation of tensors rather then their conceptual meaning. The essential content of a tensor (at a point) is its multilinearity.
The terms "covariant" and "contravariant" can have different meanings in the same context depending on their usage. For example: A little mentioned notion is that a single vector can have covariant and contravariant components. For details please seeE.g. a vector and a covector at a point both look like an n tuple of numbers, but when you change coordinates one changes by the transpose of the matrix changing the other.
Of course conceptually they differ even at a point, as one is a tangent vector and one is a linear form acting on tangent vectors.
Do you buy any of this?
Why do you call the summation a second rank tensor? It is not. gjk is a tensor of rank two. The differentials dxk are tensors of rank one. The summation is a contraction of a tensor of rank two with two tensors of rank one giving a tensor of rank zero.mathwonk said:An equation like summation gjk dxj dxk, as on the site you referenced, is a covariant tensor of second rank, because it is a second degree homogeneous polynomial in the expressions dxj, dxk, which are themselves covariant tensors of first rank.
No. They are not multiplied together. They are summed. That is a huge difference.i.e. it is of rank 2, because there are two of them multiplied together.
I recommend learning how to use subscripts and superscripts on this forum. That way I can tell if you're using them or not. I don't see why you're refering to the differentials as a basis. A basis is a vector and not a component like dxj.if we consider only one tangent space isomorphic to R^n, its dual has basis dx1,...dxn, and the second tensor product of the duals has basis dxjdxk, for all j,k, (where the order matters).
From here onwards we shall adopt a much used convention which is to confuse a tensor with its components. This allows us to refer simply to the tensor Tab, rather than the tensor with components Tab.
g is literally the tensor while gab is literally the components. They are defined by gab = g(ea,eb).
The components of tensors are in italics.mathwonk said:In that same spirit, on the site
the symbols dx^j in equation (1) should be bold, since they are the entirely analogous basic covariant 1-tensors.
That expression is a tensor of rank 0. If you notice, it is the contraction of a second rank covariant tensor with two rank 1 tensors. Such a contraction is always a tensor of rank zero. Why do you keep calling the interval a dstensor?The fact that they are not bold, leads to the confusion that this expression denotes a 0 - tensor instead of a 2 - tensor,
Nope. Sorry.i.e. equation (1) is a sum of, scalar multiples of, pairwise products of, basic 1 tensors,
hence it is a homogeneous "polynomial" of degree 2 in the basic 1 tensors,
i.e. a 2-tensor.
Does that seem believable?
Yep. I've seen them and have read part of them several years ago and during the last few years as reference. I've also read MTW as well as many other GR and tensor analysis texts.mathwonk said:Pete,
Here is a good looking reference for notes on relativity that uses both indices and the conceptual approach, by a clear expert.