Presentation on theme: "One Form. Scalar Field Map Goes from manifold M RGoes from manifold M R All points in M go to RAll points in M go to R Smooth, i.e. differentiableSmooth,"— Presentation transcript:
Scalar Field Map Goes from manifold M RGoes from manifold M R All points in M go to RAll points in M go to R Smooth, i.e. differentiableSmooth, i.e. differentiable Function f(x, y) Map U R Open set U M Region U is diffeomorphic to E 2 (or E n )
One-Form The scalar field is differentiable. Expressed in local variablesExpressed in local variables f associated with a chartf associated with a chart Need knowledge of the coordinates from the local chart ffrom the local chart f short cut is to use .short cut is to use . The entity d is an example of a one-form.
Operator and Coordinates The derivative of the one- form can be written as an operator. Chain rule applied to x, yChain rule applied to x, y A point can be described with other coordinates. Partial derivatives affected by chain rulePartial derivatives affected by chain rule Write with constants reflecting a transformationWrite with constants reflecting a transformation
Partial Derivative The partial derivatives point along coordinate lines. Not the same as the coordinates.Not the same as the coordinates. y = const. x = const. y x Y X Y = const. X = const.
Vector Field General form of differential operator: Smooth functions A, B Independent of coordinate Different functions a, b Transition between charts This operator is a vector field. Acts on a scalar field Measures change in a direction p p’ (p’) (p)(p)
Inner Product The one-form carries information about a scalar field. Components for the termsComponents for the terms The vector field describes how a scalar field changes. The inner product gives a specific scalar value. Express with componentsExpress with components Or withoutOr without
Three Laws Associativity of addition Associativity of multiplication Identity of a constant k = const.
Dual Spaces Vector field on Q Contravariant vectors Components with superscriptsComponents with superscripts Transformation rule:Transformation rule: One form on Q Covariant vectors Components with subscripts next