Divergence and curl

Divergence measures local flux expansion; curl measures local circulation in three dimensions.

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Green, Stokes and divergence theorems
Under smoothness and orientation hypotheses, boundary integrals equal corresponding interior derivative integrals.
Line integrals
A line integral accumulates a scalar density or vector-field work along a curve.
Antiderivatives
An antiderivative has the given function as its derivative; on an interval, any two antiderivatives differ by a constant.
Chain rule
The derivative of a differentiable composition combines the derivative of the outer function with that of the inner function.
Change of variables
A suitable coordinate change transforms volume elements through the absolute Jacobian determinant.