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All three of the Pauli matrices can be compacted into a single expression: = (+), where the solution to i 2 = −1 is the "imaginary unit", and δ jk is the Kronecker delta, which equals +1 if j = k and 0 otherwise.
Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with ~, while those unmarked with such will be associated with .)
The sum of the entries along the main diagonal (the trace), plus one, equals 4 − 4(x 2 + y 2 + z 2), which is 4w 2. Thus we can write the trace itself as 2 w 2 + 2 w 2 − 1 ; and from the previous version of the matrix we see that the diagonal entries themselves have the same form: 2 x 2 + 2 w 2 − 1 , 2 y 2 + 2 w 2 − 1 , and 2 z 2 + 2 w ...
The collection of tensors on a vector space and its dual forms a tensor algebra, which allows products of arbitrary tensors. Simple applications of tensors of order 2 , which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product ...
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 derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. [1]
Let r(x) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r(x). At each point we can construct a small line element dx. The square of the length of the line element is the scalar product dx • dx and is called the metric of the space.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.