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.
Temperature and other scalars are quantities without direction, have no components, and are zero-order tensors {order, tensor}. Motion, momentum, force, and other vectors have one direction and are first-order tensors. Metric and other matrices can represent two-dimension interactions and are second-order tensors. Vector curvatures can represent four-dimension interactions and are fourth-order tensors.
Curvature tensor is 0 if there is no torsion {first Bianchi identity, symmetry} {Bianchi symmetry, tensor}.
Curvature-tensor derivative is 0 if there is no torsion {Bianchi identity, tensor} {second Bianchi identity, tensor}.
Coordinates can depend directly on dimensions {contravariant, tensor}. Coefficients a times basis vectors e result in coordinates x: a(i) * e(i) = x(i).
covariant
Covariant components relate to contravariant components. If basis vectors are orthogonal, covariant components and contravariant components are equal: a(i) = x(i) / e(i).
If basis vectors are curved coordinates, then a(i) = g(i,j) * a(j), where g(i,j) depend on basis vectors e(i) and e(j). Some g(i,j) components are for covariance, some for contravariance, and some for both. g(i,j) tensor relates basis vectors. g(i,j) elements are functions of curved-space positions.
Euclidean space
g(i,j) elements are 1 or 0 for flat space with orthogonal basis vectors.
Physical quantities or coordinates can transform from one coordinate system to another {coordinate transformation}. First coordinate-system vector components are linear functions of second coordinate-system components.
tensor
Tensor coefficients are weights by which to multiply old variables to get new variables. Tensor-term number is old-component number times new-component number. Scalar product of outer-product tensor and old basis vectors obtains new basis-vector scalars.
projection
Linear transformation projects old onto new. Linear transformation is affine geometry.
contravariant
Contravariant component {contravariant}, such as dx, multiplies with tensor. Terms with same index, such as ii, have coefficient one. Terms with different indexes, such as ij, have coefficient zero. In contravariant transformation, only diagonal terms remain. Contravariant component sum is vector expressed in old basis vectors. Diagonal terms are scalars for new basis vectors.
covariant
Covariant component {covariant}, such as partial derivative, is contravariant component times tensor. Terms with different indexes, such as ij, have coefficient one. Terms with same index, such as ii, have coefficient zero. In covariant transformation, diagonal terms are not present. Covariant component sum is weight matrix. Non-diagonal terms are weights.
covariance and contravariance
Covariant means that different old and new components interact. Contravariant means that same old and new components interact. Together, they account for all interactions. Contravariant reduces dimension by one. Covariant does not change dimensions. If components are orthogonal, as in Euclidean space, covariant and contravariant components are the same. Contravariant or covariant transformation does not change symmetrical-tensor value. Contravariant or covariant transformation only changes sign of odd number of skew-symmetrical tensor transformations.
Linear functions can find coefficients of scalar products from original variables and basis vectors {covariant, tensor}: a(i) = x(i) * e(i). Covariant components relate to contravariant components by relations between basis vectors. If basis vectors are orthogonal, covariant components and contravariant components are equal. If basis vectors are curved coordinates, then a(i) = g(i,j) * a(j), where g(i,j) depend on basis vectors e(i) ... e(j). Some g(i,j) components are for covariance, some for contravariance, and some for both. g(i,j) tensor relates basis vectors. g(i,j) elements are functions of curved-space positions.
g(i,j) elements are 1 or 0 for flat space with orthogonal basis vectors.
covariant transformation
Terms with different indexes, such as ij, have coefficient one. In covariant transformation, new coefficients are new-vector coefficients, and variable number stays the same.
Notation conventions {Einstein summation convention} can denote tensors.
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.
Tensor order can reduce by two {tensor contraction}. If tensor has contravariant component and covariant component, sum tensor over each component, at same time, to eliminate each component.
Scalar products contract second-order tensors and are sums of squares along diagonal {trace, tensor} {spur, tensor}.
Volumes {volume, tensor} are determinants of third-order skew-symmetrical covariant tensors.
Space has dimensions. Vectors {basis vector} can lie along coordinate axes. Other vectors are basis-vector linear combinations.
contravariant
For point, coefficients a(i) times basis vectors e(i) result in point coordinates x(i): a(i) * e(i) = x(i), where i is number of dimensions. Points can move. New coefficients x(i) of same dimensions e(i) relate to old coefficients a(i): a(i) * e(i) = x(i).
covariant
For points, coordinate system can change. New-dimension coefficients a(i) relate to old dimensions e(i), because new dimensions are linear transformations x(i) of old dimensions: a(i) = x(i) * e(i).
relation
Covariant components relate to contravariant components. If basis vectors are orthogonal, covariant components and contravariant components are equal. If basis vectors are curved coordinates, a(i) = g(i,j) * a(j), where a(i) and a(j) are coefficients and g(i,j) is tensor relating basis vectors e(i) ... e(j). Some g(i,j) components are for covariance, some for contravariance, and some for both. g(i,j) elements are functions of curved-space positions. g(i,j) elements are 1 or 0 for flat space with orthogonal basis vectors.
Tensor can express coordinates {coordinate system}. For example, space points can be vectors: a*i + b*j + c*k. Coordinates can be perpendicular or not perpendicular. Physical quantity depends on coordinates and can use tensor form. For example, momentum can be vector: mass * (a*i + b*j + c*k).
Surface can deviate from flatness {curvature, tensor}.
local
Curvature is at point and is local property.
intrinsic
Curvature is intrinsic to surface and does not depend on outside reference points.
curve
For curves, curvature at point is curvature-radius reciprocal.
surface
For surface points, curvature R is product of reciprocals of maximum curvature radius R1 and minimum curvature radius R2: R = (1 / R1) * (1 / R2). Rotating plane around normal to surface can find both.
surface: triangle
Triangles are three geodesics. Curvature is triangle-angle sum minus pi radians, all divided by triangle area: r = (angle sum - pi) / area.
surface: sphere
Surface curvature is area by surface projection onto a sphere, divided by surface area: r = (projection area) / area.
surface: normals
Curvature is solid angle in radians by normals to surface over surface region, divided by surface area.
surface: volume
Volume can also find curvature.
surface: hexagon
To measure curvature around point, use regular hexagon around the point and measure angles.
bending
If surface bends without stretching, curvature stays constant, because one radius increases and other radius decreases proportionally.
space-time: curvature tensor
In space-time, geodesic deviations, measured along each dimension, are metric second derivative and are second-order differential forms. The result is fourth-order tensor whose matrix coefficients have one covariant and three contravariant indices. For four-dimensional space-time, tensor has 20 independent terms. Metric coefficients are potentials.
Tensor is skew-symmetric and cyclic symmetric and is commutative, with diagonal terms equal zero.
space-time: Weyl tensor
Tensors {Weyl tensor} can measure gravity-field tidal distortion.
space-time: Riemann curvature tensor
At points, total space curvature depends on Ricci tensors and Weyl tensors. Curved space can be Euclidean space {Riemannian manifold} locally. Curved space-time can be Minkowski space {Lorentzian manifold} {pseudo-Riemannian manifold} {semi-Riemannian manifold} locally.
space-time: distance curvature
Second-order tensor {distance curvature} can derive from vector curvature, by contracting matrix coefficients.
space-time: scalar curvature
Tensor trace is scalar invariant. First-order tensors {scalar curvature} can derive from distance curvatures, by contracting to put metric-coefficient second derivatives into linear forms. Scalar curvature is the only such invariant. Physical world has four dimensions, because only such manifold results in curvature invariance.
Functions have functions {form, mathematics}.
types
Functions are 0-form manifolds.
Exterior derivatives of functions with basis vectors are 1-forms {differential form} {covector} {covariant vector} {1-form, basis}. Tensors operate on 1-forms to give real numbers. Linear operations on vectors can give integral numbers of equally spaced phase surfaces, of de-Broglie particle waves, through which vector passes. Force, energy, momentum, and velocity directional derivatives are 1-form gradients. Anticommutative tensor products and wedge products of two 1-forms are 2-forms. Anticommutative wedge products of three 1-forms are 3-forms. Antisymmetric covariant tensors are k-forms.
algebra
Exterior-derivative wedge products form exterior algebra (Elie Cartan).
dual
p-forms and (n-p)-forms are duals on n-manifolds {Hodge star}. 1-form and vector field are duals and interact to make scalar products. Tangent vector has covector dual.
Two manifold points have shortest path {geodesic, tensor}| between them. Geodesic is straightest possible direction between two points.
metric
Quadratic differential linear metric forms can measure geodesic length: ds^2. Geodesic length is sum from points i = 1 to i = m, and from points j = 1 to j = m, of g(i, j) * du(i) * du(j). Coefficients g(i, j) = Du(i) / Du(j), where D are partial derivatives and u are coordinates.
geometry
Geodesic metric defines surface geometry at manifold points.
linear
Using only local operations allows geodesic to be linear.
operator
Geodesic metric operates on vectors to give squared lengths. Squared length can be greater than zero {space-like vector}, less than zero {time-like vector}, or equal to zero {light-like vector}.
space-time
In four-dimensional space-time, particles move along maximum spatial-length lines, which is the shortest possible time as measured in particle reference frame. In flat space-time, geodesics are straight lines. On spheres, geodesics are on great circles.
For surface, quadratic differential linear form can measure geodesic length {metric, length}|: ds^2.
Four-dimensional space has vector curvature with 4x4x4x4 terms, which are linear function-second-derivative combinations. However, terms are in pairs, so four-dimensional vector curvature can contract to 16 independent terms in 4x4 matrices {Ricci tensor}. The other four terms are vector components. Ricci tensor measures volume change, as gravity causes space to contract, and equals energy-momentum tensor. Ricci tensor equals mass-energy density, which is pressure.
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Date Modified: 2022.0225