What Are Tensors and Why Are They Important in Physics?

[WannabeNewton]

But then how is the definition coordinate free? You are forced to choose some coordinates in order to work with tensors.

This depends on what you mean by "work" and in what context but the definition is clearly coordinate-free. I would suggest you take a look at Roman's linear algebra text.

 

[Jhenrique]

Will be the Laplacian, $$\bigtriangledown ^2$$, in actually, a Tensor?? Will be a simplification of Hessian tensor?

 

[Jhenrique]

Come think in 2D. Given 2 values, a and b, it's possible to associate a point P with coordinates (a, b) and trace a straight between the origem and the point P. Ready, we have a vector! So, analogously, given 4 values (a, b, c, d), should be possible to associate these values with straight, and, somehow associate a plane with this straight. This, geometrically, would be a tensor. I think...

 

[PSarkar]

This depends on what you mean by "work" and in what context but the definition is clearly coordinate-free. I would suggest you take a look at Roman's linear algebra text.

Can you explain why the multilinear array definition is coordinate free as this is not obvious to me. If you need to choose coordinates to get the tensor, then the tensor depends on which coordinates you choose. You definitely cannot write one down without choosing coordinates. This definition misses the essence of tensor which is independent of any coordinate and so we should have a definition and representation without coordinates.

I think jgens answer about collecting all the coordinate representation of the tensors together make sense since it will no longer depend on the coordinates. But the down side is you still can't write one down without coordinates.

 

[WannabeNewton]

Can you explain why the multilinear array definition is coordinate free as this is not obvious to me.

Sorry when you said "the definition" I thought you were referring to the usual algebraic definition of a tensor as this is what I take to be "the definition" of a tensor. I find it hard to look at anything relying on coordinate representations as a definition hence I agree that a definition in terms of coordinate components is archaic and inferior for the reasons stated by you and others.