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One Form

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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 )

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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.

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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

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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.

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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)

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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

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

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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

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2D case: If or then 2 or 3D cases: Several dimensions: U(x,y,z) Compact notation using vector del, or nabla: Another notation: Partial derivative is.

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