Exterior

In mathematics, the exterior derivative operator of differential geometry extends the concept of the differential of a function to differential forms of higher degree. more...

Home
ATV Parts
Apparel & Merchandise
Automotive Tools
Aviation Parts
Boat Parts
Car & Truck Parts
Car Audio, Video
Manuals & Literature
Motorcycle Parts
Other Vehicle Parts
Personal Watercraft Parts
Racing Parts
Services & Installation
Snowmobile Parts
Vintage Car & Truck Parts
AC & Heating
Accessories
Air Intake & Fuel Delivery
Brakes
Charging & Starting Systems
Cooling System
Decals, Emblems, Detailing
Engines & Components
Exhaust
Exterior
Antennas
Bumpers
Car Covers
Doors & Door Handles
Fenders
Grilles
Hoods
Locks & Hardware
Mirrors
Mouldings & Trim
Nerf Bars & Running Boards
Other
Racks
Spoilers & Wings
Sunroof, Convertible &...
Tailgates & Liftgates
Truck Bed Accessories
Trunk Lids & Parts
Windshield Wiper Systems
Filters
Gaskets
Gauges
Glass
Ignition Systems
Interior
Lighting & Lamps
Other Parts
Radio & Speaker Systems
Safety & Security
Salvage Parts Cars
Suspension & Steering
Transmission & Drivetrain
Wheels, Tires, & Hub Caps
Wholesale Lots

It is important in the theory of integration on manifolds, and is the differential (coboundary) used to define de Rham and Alexander-Spanier cohomology. Its current form was invented by Élie Cartan.

Definition

The exterior derivative of a differential form of degree k is a differential form of degree k + 1.

For a k-form ω = f_I dx_I over Rⁿ, the definition is as follows:

For general k-forms Σ_I f_I dx_I (where the multi-index I runs over all ordered subsets of {1, ..., n} of cardinality k), we just extend linearly. Note that if i = I above then

Properties

Exterior differentiation satisfies three important properties:

linearity;

the wedge product rule (see antiderivation);

and d² = 0, a formula encoding the equality of mixed partial derivatives, so that always;

It can be shown that the exterior derivative is uniquely determined by these properties and its agreement with the differential on 0-forms (functions).

The kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials).

The exterior derivative is natural. If f: M → N is a smooth map and Ω^k and Ω^k+1 are the contravariant smooth functors that assign correspondingly to each manifold the space of k- and k+1-forms on the manifold, then the following diagram commutes

so d(f*ω) = f*dω, where f* denotes the pullback of f. Thus d is a natural transformation from Ω^k to Ω^k+1.

Invariant formula

Given a k-form ω and arbitrary smooth vector fields V₀,V₁, …, V_k we have

Read more at Wikipedia.org