Coordinates have unit vectors, such as u for x-axis and v for y-axis. Vector from origin can have coordinates: (2,3) = 2*u + 3*v, where u and v are unit vectors for two-dimensions. Straight vector has constant slope.
Lines and surfaces can curve. At point, line or surface has curvature. Slope change indicates curvature amount. For two dimensions, two orthogonal directions, u and v, can change slope: du and dv, where d is differential. Total curvature has coefficients that depend on dimensions and is sum of changes along dimensions: (Df(u,v) / Dv) * du + (Df(u,v) / Du) * dv, where D is partial derivative and d is differential.
linear
Linear functions depend on one variable raised only to first power: C * x. Bilinear functions depend on product of first-power variables: C * x * y.
symmetry
Functions have symmetry if function variables can interchange, for example, they are Euclidean space dimensions.
tensor
Tensors are linear functions of coefficients times any number of variables: c * v1 * v2 * ... * vN, where c is coefficient, vi are variables, and N can be infinite. Tensors can be sums of these terms: c1 * i1 * j1 * ... * n1 + c2 * i2 * j2 * ... * n2 + ... + cN * iN * jN * ... * nN, where n can be infinite. Vectors are tensors: Cu * u or Cu * u + Cv * v.
scalar product
Bilinear forms are tensors: Cuv * du * dv. Bilinear tensors can be sum over all ij of g(ij) * du * dv, where i and j are dimensions, g is function of dimensions, and u and v are dimension unit vectors. Scalar product of two vectors (a,b) and (c,d) is a*c*i*i + b*d*j*j + a*d*i*j + b*c*j*i = a*c + b*d, which is symmetric bilinear tensor. In scalar products, terms with same index, such as ii, have coefficient one, and terms with different indexes, such as ij, have coefficient zero {contravariant transformation}. Vector projection onto itself makes 100% = 1 of vector. Vector projection onto perpendicular makes zero.
scalar product: symmetry
Tensor projection and scalar product are symmetric, because answer is the same if either vector projects onto other vector.
scalar product: projection
Scalar product projects one vector onto another to find length. Tensor transformations can project {projection, tensor} one vector onto another vector, to give length.
quadratic
If two vectors are the same, scalar product is quadratic. For vector (a,b), sum over all ij of g(ij) * du * du = a*a*i*i + b*b*j*j = a^2 + b^2.
cross product
Vector cross products are vectors and tensors: (a,b) and (c,d) make a*c*i*i + b*d*j*j + a*d*i*j + b*c*j*i = (a*d - b*d)*k, where j*i = - i*j = -k, because opposite direction. In cross products, terms with same index, such as ii, have coefficient zero, and terms with different indexes, such as ij, have coefficient one {covariant transformation}. Divergence of vector from itself is zero. Divergence of vector onto perpendicular makes 100% = 1 divergence. Tensors can be scalar, vector, matrix, and tensor products.
Mathematical Sciences>Calculus>Vector>Tensor>Operations
3-Calculus-Vector-Tensor-Operations
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224