Linear forms {tensor, mathematics} have dimension or variable coefficients. Scalars, vectors, and matrices are tensors. Vectors are rank-1 covariant tensors.
purposes
Tensors can transform coordinates. Tensors can sum over all component combinations or any component. Tensors describe vector operations, complex numbers, analytic geometry, and differential geometry. All physical laws are tensor relations. For example, tensors describe flow, crystal deformations, and elasticity. All intensive physical quantities can be tensors. Tensors can measure extensive quantities, such as mass, momentum, energy, inertia moment, length, area, and volume.
differentiation
To differentiate tensor, raise tensor order by one. Differentiating first-order tensor results in second-order tensor. Differentiating tensor makes tensor gradient.
integration
To integrate tensor, lower tensor order by one, by summing over one component.
area
Tensor transformations find surface areas, which are outer products. Tensor determinants give areas. Transformations can be second-order skew-symmetrical covariant tensors or skew-symmetric bilinear forms: sum over all ij of g(ij) * du *dv. Terms with same index, such as ii, have coefficient zero. Terms with different indexes, such as ij, have coefficient one. In covariant transformations, new coefficients are new-vector coefficients, and variable number stays the same.
invariants
Tensor invariants are distance, curvature, sum of curve partial-derivative squares, sum of curve second derivatives, sum of area second derivatives, and sum of volume second derivatives. Tensor invariants have both contravariant and covariant components. They can contract.
tensor density
Tensors can multiply metric-coefficient-determinant square roots. Tensor densities are contravariant and symmetric, like divergence. If vectors have same direction, metric-coefficient-determinant square root is zero.
polynomials
Many-variable polynomials can be equivalent to tensors {polynomial, tensor} {tensor, polynomial}. For example, double sums with two-variable terms have quadratic form a*x*x + b*x*y + c*y*y, which can be equivalent to scalar products. Differential forms can use dx instead of x.
Mathematical Sciences>Calculus>Vector>Tensor
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Date Modified: 2022.0224