Differential form

A differential form is a smooth field of alternating covariant tensors on a manifold. Its degree determines the dimension of geometric objects over which it can be integrated.

Cotangent space: The vector space of linear functionals on a tangent space at a point. A differential 1-form assigns a covector to each point of a manifold.
Exterior derivative: An operator that maps a differential k-form to a differential (k+1)-form and satisfies a graded product rule. It captures generalized differentiation while always obeying d squared equals zero.
Line integral: An integral of a scalar or vector quantity along a curve. Integrating a 1-form along a curve generalizes work and circulation integrals.
Differential 0-form: A smooth real-valued function on a manifold, regarded as a degree-zero differential form. It is the simplest form and shows that forms include ordinary smooth functions.
Élie Cartan: A French mathematician who developed exterior differential calculus and major ideas in differential geometry. Cartan established the systematic calculus that treats forms and exterior differentiation as central tools.
Exterior algebra: An algebra built from alternating products of vectors or covectors. Its graded spaces provide the algebraic home for differential forms.
Pullback: An operation that transfers covariant tensors from a target space to a source through a smooth map. Forms can be transported along maps without choosing coordinates or a metric.
Flux: The amount of a vector field passing through an oriented surface per unit time or area, depending on context. A 2-form can encode oriented flux through surfaces.
Differential 1-form: A smooth field of covectors on a manifold. It is the degree-one case, acting on one tangent vector at a time.
Hermann Grassmann: A German mathematician whose work developed an algebra of extension and ideas foundational to exterior algebra. His algebraic ideas underpin the alternating products used to build forms.

Linked from 34 pages