How do you work out a tensor product?

Tensor product distributes over addition. That is, (a+b)⊗c=a⊗c+b⊗canda⊗(b+c)=a⊗b+a⊗c. Furthermore, for any real scalar λ, we have (λa)⊗b=a⊗(λb)=λ(a⊗b).

How do you describe a tensor?

Tensors are simply mathematical objects that can be used to describe physical properties, just like scalars and vectors. In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor, and a vector is a first rank tensor.

What is the difference between tensor product and outer product?

In linear algebra, the outer product of two coordinate vectors is a matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra.

How do you read tensor notation?

In the most general representation, a tensor is denoted by a symbol followed by a collection of subscripts, e.g. In most instances it is assumed that the problem takes place in three dimensions and clause (j = 1,2,3) indicating the range of the index is omitted.

What is tensor give example?

A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge. Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor.

What is the purpose of the tensor product?

Roughly, the purpose of the tensor product, $\\otimes$, is to make the following statement true: $$ ext{functions}(X imes Y) = ext{functions}(X)\\otimes ext{functions}(Y)$$ The specific details about which spaces of functions to choose depend on the type of mathematical object you are interested in.

How do you find the tensor product of two vectors?

The first is a vector (v,w) ( v, w) in the direct sum V ⊕W V ⊕ W (this is the same as their direct product V ×W V × W ); the second is a vector v ⊗w v ⊗ w in the tensor product V ⊗W V ⊗ W. And that’s it! Forming the tensor product v⊗w v ⊗ w of two vectors is a lot like forming the Cartesian product of two sets X×Y X × Y.

What is the difference between ordered pair and tensor product?

Essentially the difference between a tensor product of two vectors and an ordered pair of vectors is that if one vector is multiplied by a nonzero scalar and the other is multiplied by the reciprocal of that scalar, the result is a different ordered pair of vectors, but the same tensor product of two vectors.

What is a covariant tensor?

Every quantity which under a coordinate transformation, transforms like the derivatives of a scalar is called a covariant tensor. Accordingly, a reasonable generalization is having a quantity which transforms like the product of the components of two contravariant tensors, that is