Comment on number box o number box
Dadifer@lemmy.world 7 months ago
A tensor is a special box of numbers that doesn’t change under coordinate transformation.
Where does this definition come from?
All the geometric definitions of tensors I have met always assumed a base, such that a change of coordinate or of parametrization would change the values of the tensor. Unless you define the tensor by its action instead of its values?
That’s exactly correct. It’s similar to how a vector in R^2 is just an arrow with a magnitude and a direction. When you represent that arrow in different bases, the arrow itself isn’t changing, just the list of numbers you use to represent them. Likewise, tensors do not change when you change bases, but their representations as n dimensional grids of numbers do change.
Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
Isn’t a tensor the generalization of scalar, vector matrix and so on? (PLUS the invariance under coordinate transforms?)
A box would be 3-dimensional indicating that tensors have 3 indices when in reality they have n-indices. Ir am i reading it wrong?
It’s an n-dimensional box
kogasa@programming.dev 7 months ago
Which is really a roundabout way of saying a tensor is a multilinear relationship between arbitrary products of vectors and covectors. They’re inherently geometric objects that don’t depend on a choice of coordinate system. The box of numbers is just one way of looking at a tensor, like a matrix is to a linear transformation on a vector space