ExteriorDerivative(omega)

d(alpha)(X, Y) = X(alpha(Y)) - Y(alpha(X)) - alpha([X, Y]),

d(beta)(X, Y, Z) = X(beta(Y, Z)) - Y(beta(X, Z)) + Z(beta(X, Y)) - X(beta([Y, Z])) + Y(beta([X, Z ])) - Z(beta([X, Y])),

where X, Y, Z are vector fields.  Most of the references listed on the DifferentialGeometry References page contain the general formula for the exterior derivative of a p-form.

      [i]  for functions f, d(f)(X) = X(f),  where X  is any vector field;

      [ii]  d(alpha &w beta) = d(alpha) &w beta + (- 1)^p alpha &w d(beta), where alpha and beta are differential forms and p is the degree of alpha; and

      [iii] d(d(alpha)) = 0.

The explicit coordinate formulas for the exterior derivatives of a function, a 1-form and a 2-form in 3 dimensions are given in Example 1.

Example 1.

We initialize a 3-dimensional manifold with coordinates [x, y, z].

We use the declare command in  PDEtools to display the partial derivatives of the functions a(x, y, z), b(x, y, z) and c(x, y, z) in compact form.

The exterior derivative of a function:

The exterior derivative of a 1-form:

The exterior derivative of a 2-form:

Example 2.

By way of an example, we illustrate the fact that d^2 = 0.

Example 3.

The ExteriorDerivative command can also be applied to a list of forms or a matrix of forms.

Example 4.

The ExteriorDerivative command can also be used with adapted frames.  First we define an adapted coframe for M.

Example 5.

The ExteriorDerivative command can be used with Lie algebras.

Example 6. 

The ExteriorDerivative command can also be used with abstract differential forms.    

DifferentialGeometry, LieBracket, DeRhamHomotopy, PDEtools[declare]