differential form

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.

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3-Calculus-Vector-Tensor-Kinds

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Date Modified: 2022.0224