total differential

The exterior derivative of a function f is the one-form d f = sum_i (df)/(dx_i) d x_i written in a coordinate chart (x_1, ..., x_n). Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k-forms using the formula d(α⋀β) = d α⋀β + (-1)^k α⋀d β, when α is a k-form and where ⋀ is the wedge product. The exterior derivative of a k-form is a (k + 1)-form.

differential k-form | exterior algebra | Hodge star | Jacobian | manifold | Poincaré's lemma | Stokes' theorem | tangent bundle | tensor | wedge product