Tensor product

In mathematics, the tensor product



V
⊗
W


{\displaystyle V\otimes W}
of two vector spaces V and W (over the same field) is a vector space which can be thought of as the space of all tensors that can be built from vectors from its constituent spaces using an additional operation which can be considered as a generalization and abstraction of the outer product. Because of the connection with tensors, which are the elements of a tensor product, tensor products find uses in many areas of application including in physics and engineering, though the full theoretical mechanics of them described below may not be commonly cited there. For example, in general relativity, the gravitational field is described through the metric tensor, which is a field (in the sense of physics) of tensors, one at each point in the space-time manifold, and each of which lives in the tensor self-product of tangent spaces




T

x


M


{\displaystyle T_{x}M}
at its point of residence on the manifold (such a collection of tensor products attached to another space is called a tensor bundle).

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