(i.e., differentiability class ) is always the zero vector: It can be easily proved by expressing in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex. Curl of curl Here ∇2 is the vector Laplacian operating on the vector field A. Curl of divergence is not defined The divergence of a vector field A is a scalar, and you cannot take curl of a scalar quantity. Therefore A mnemonic The figure to the right is a mnemonic for some of these identities. The abbreviations used are: D: divergence, C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist. Summary of important identities Differentiation Gradient Divergence Curl Vector dot Del Operator Second derivatives (scalar Laplacian) (vector Laplacian) (Green's vector identity) Third derivatives Integration Below, the curly symbol ∂ means "boundary of" a surface or solid. Surface–volume integrals In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional boundary S = ∂V (a closed surface): (divergence theorem) (Green's first identity) (Green's second identity) (integration by parts) (integration by parts) (integration by parts) Curve–surface integrals In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve): (Stokes' theorem) Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral): Endpoint-curve integrals In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points and integration along P is from to : (Gradient theorem). See also References Further reading Mathematical identities Mathematics-related lists Vector calculus eo:Vektoraj identoj zh:向量恆等式列表