Exterior derivative

The exterior derivative is a linear operator that maps differential forms of degree k to forms of degree k+1. It generalizes the gradient, curl, and divergence while satisfying a coordinate-independent product rule and d² = 0.

Differential form: A differential form is a smooth field of alternating multilinear functions that can be integrated over oriented surfaces. The exterior derivative acts on forms, raising their degree by one.
Tangent space: The tangent space at a point is the vector space of directions in which curves can pass through that point. Differential forms evaluate tangent vectors, giving the exterior derivative its pointwise inputs.
Gradient: The gradient of a scalar field is a vector or covector encoding its direction of greatest increase. On functions, the exterior derivative produces the differential that corresponds to the gradient.
Covariant derivative: A covariant derivative differentiates tensor fields while accounting for how tangent spaces vary across a manifold. Unlike the exterior derivative, it depends on a connection and acts on general tensor fields.
Wedge product: The wedge product is an antisymmetric product that combines differential forms into a form of higher degree. Its graded product rule governs how the exterior derivative acts on products of forms.
Cotangent space: The cotangent space at a point is the dual vector space to its tangent space. Differential forms are built from cotangent vectors and their alternating products.
Curl: The curl of a vector field measures its local rotation in three-dimensional Euclidean space. In three dimensions, the exterior derivative of a 1-form corresponds to the curl.
Lie derivative: The Lie derivative measures how a tensor field changes along the flow generated by a vector field. It describes change along a flow, whereas the exterior derivative raises form degree without choosing a flow.
Stokes' theorem: Stokes' theorem equates the integral of a differential's derivative over a region with its integral over the boundary. It turns the exterior derivative into the local operator underlying a broad integral theorem.
Smooth manifold: A smooth manifold is a space locally described by smooth coordinate charts compatible on overlaps. The exterior derivative is defined on forms over smooth manifolds without depending on a chosen chart.

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