A tensor itself is a linear combination of let’s say generic tensors of the form . In the case of one doesn’t speak of tensors, but of vectors instead, although strictly speaking they would be called monads.
Every tensor is associated with a linear map that produces a scalar. For instance, a vector can be identified with a map that takes in another vector (in the presence of an inner product) and produces a scalar.
A tensor extends the notion of a matrix analogous to how a vector extends the notion of a scalar and a matrix extends the notion of a vector. A tensor can have any number of dimensions, each with its own size. A $3$ -dimensional tensor can be visualized as a stack of matrices, or a cuboid of numbers having any width, length, and height.
Tensor : Multidimensional array :: Linear transformation : Matrix. The short of it is, tensors and multidimensional arrays are different types of object; the first is a type of function, the second is a data structure suitable for representing a tensor in a coordinate system. In the sense you're asking, mathematicians usually define a "tensor" to be a multilinear function: a function of ...
I'm an electrical engineer, and thus don't often interact with the types of mathematics that involve tensors. But when I try to get a deeper understanding of certain things that I do interact with, I
What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
Regarding why a $ (0,1)$ tensor can be considered a vector, that is because (for finite-dimensional vector spaces) any vector space is isomorphic to its double dual vector space. And it seems like, using your definition, a $ (0,1)$ tensor corresponds to an element of the double dual space. See the question an answers for much better explanation than what I am giving here: math.stackexchange ...
tensor - In new latin tensor means "that which stretches". The mathematical object is so named because an early application of tensors was the study of materials stretching under tension.
The discussion focuses on the formulation of the displacement gradient tensor H in continuum mechanics, specifically its decomposition into the infinitesimal strain tensor epsilon and the rigid body rotation tensor Omega.
Mathematically, the EFE relate the Ricci curvature tensor , the metric tensor , and the stress-energy tensor , incorporating the Einstein constant . Fast Facts Einstein’s Big Reveal: Albert Einstein first presented the Einstein Field Equations in 1915.
