Sage-Tips

How many components does a tensor have?

If you want to work with 4-dimensional spaces, like in General Relativity, then tensors can have 1, 4, 16, components, and so on. It doesn’t always have nine components. It could have 3 (a vector, which is a tensor of rank 1), or 1 (a scalar, which is a tensor of rank 0), or 27 (a tensor of rank 3).

What are the components of a tensor?

The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {ei} for V and its dual basis {εj}, i.e. Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor.

How many components is a rank 3 tensor?

In 3 dimensions, a totally antisymmetric (rank three) tensor has one component.

How many components does a rank 2 tensor in 3d space have?

A tensor of rank 2 is just a matrix. If your underlying vector space is of dimension 3, as you seem to assume, then this would be a 3×3-matrix, which obviously has 9 components.

How many independent components are there in a symmetric tensor?

Now, for each of these 6 combinations there are 4(4+1)2=10 independent combinations of α and β, as the tensor is symmetric under the exchange of these two indices. Thus, there are in total 6×10=60 independent components of the tensor.

How many components are present in a second order tensor quantity?

Like vector components, second-order tensor components change with a change in coordinate system. Once a coordinate system is chosen, the nine components of a second-order tensor can be represented by a 3 × 3 matrix, or by an italic symbol having two indices, such as T i j for the stress tensor.

What is stress tensor in engineering?

The Stress Tensor Stress is defined as force per unit area. If we take a cube of material and subject it to an arbitrary load we can measure the stress on it in various directions (figure 4). These measurements will form a second rank tensor; the stress tensor.

What is the difference between tensors and tensors?

While tensors are defined independent of any basis, the literature on physics often refers to them by their components in a basis related to a particular coordinate system. An elementary example of mapping, describable as a tensor, is the dot product, which maps two vectors to a scalar.

How do the components of a tensor change under a transformation?

Just as the components of a vector change when we change the basis of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis.

Can a tensor be represented as a multidimensional array?

A tensor may be represented as a (potentially multidimensional) array. Just as a vector in an n – dimensional space is represented by a one-dimensional array with n components with respect to a given basis, any tensor with respect to a basis is represented by a multidimensional array.

What do the numbers n and m mean in a tensor?

The numbers of, respectively, vectors: n (contravariant indices) and dual vectors: m (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law.