The modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make references to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used heavily in abstract algebra and homological algebra, where tensors arise naturally.
Let V and W be two real vector spaces. Their tensor product is a real vector space
A tensor on the vector space V is then defined to be an element of (i.e. a vector in) the following vector space:
If there are m copies of V and n copies of V* in our product, the tensor is said to be of type (m, n) and of contravariant rank m and covariant rank n. The tensors of rank zero are just the scalars R, those of contravariant rank 1 the vectors in V, and those of covariant rank 1 the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors).
Note that the (1,1) tensors
In differential geometry, physics and engineering, we usually deal with tensor fields on differentiable manifolds. (The term "tensor" is sometimes used as a shorthand for "tensor field".) For instance, the curvature tensor is discussed in differential geometry and the stress-energy tensor is important in physics and engineering. Both of these are related by Einstein's theory of general relativity. In engineering, the underlying manifold will often be Euclidean 3-space. A tensor field assigns to any given point of the manifold a tensor in the space
For any given coordinate system we have a basis {e_{i}} for the tangent space V (note that this may vary from point-to-point if the manifold is not linear), and a corresponding dual basis {e^{i}} for the cotangent space V* (see dual space). The difference between the raised and lowered indices is there to remind us of the way the components transform.
For example purposes, then, take a tensor A in the space