1. One Form

2. 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 )

3. 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.

4. 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

5. 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.

6. 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)

7. 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

8. Three Laws  Associativity of addition  Associativity of multiplication  Identity of a constant k = const.

9. 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