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A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index.
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
The tensor product of V and its dual space is isomorphic to the space of linear maps from V to V: a dyadic tensor vf is simply the linear map sending any w in V to f(w)v. When V is Euclidean n-space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in ...
A vector's components change scale inversely to changes in scale to the reference axes, and consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to the transformation of the reference axes, (with example transformations including rotation and ...
If W is the tensor product bundle of V with L, then W is a bundle of vector spaces of just the same dimension as V. This allows one to define the concept of tensor density, a 'twisted' type of tensor field. A tensor density is the special case where L is the bundle of densities on a manifold, namely the determinant bundle of the cotangent bundle.
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
The normalized basis vectors are e r = (cos θ, sin θ), e θ = (−sin θ, cos θ) and the scale factors are h r = 1 and h θ = r. The fundamental tensor is g 11 =1, g 22 = r 2 , g 12 = g 21 =0. Line and surface integrals