How do you find the exterior derivative?

If f is a smooth function (a 0-form), then the exterior derivative of f is the differential of f . That is, df is the unique 1-form such that for every smooth vector field X, df (X) = dX f , where dX f is the directional derivative of f in the direction of X.

What is meant by differential form?

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

Why is it called exterior derivative?

For when the flux exits one end, it immediately enters the next. This is called the exterior derivative because the integral of the “exterior derivative” leaves the difference of the exterior behind. Loosely the word “integral” cancels “derivative” leaving ” exterior” behind… hence its name.

What is D in derivatives?

The symbol dydx. means the derivative of y with respect to x. If y=f(x) is a function of x, then the symbol is defined as dydx=limh→0f(x+h)−f(x)h. and this is is (again) called the derivative of y or the derivative of f. Note that it again is a function of x in this case.

What is the derivative of 2x?

2
The derivative of 2x is equal to 2 as the formula for the derivative of a straight line function f(x) = ax + b is given by f'(x) = a, where a, b are real numbers. Differentiation of 2x is calculated using the formula d(ax+b)/dx = a.

How do you find the exterior derivative of a differential form?

The exterior derivative of a differential form of degree k (also differential k -form, or just k -form for brevity here) is a differential form of degree k + 1 . If f is a smooth function (a 0 -form), then the exterior derivative of f is the differential of f .

Why is the exterior derivative used as the differential for de Rham?

Because the exterior derivative d has the property that d2 = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold.

What is the exterior derivative of a linear function?

The exterior derivative is defined to be the unique ℝ -linear mapping from k -forms to (k + 1) -forms that has the following properties: df is the differential of f for a 0 -form f. d(df) = 0 for a 0 -form f. d(α ∧ β) = dα ∧ β + (−1)p (α ∧ dβ) where α is a p -form.

What is the exterior derivative of a boundary map?

As suggested by the generalized Stokes’ theorem, the exterior derivative is the “dual” of the boundary map on singular simplices.