Leibniz Integral Rule - Three-dimensional, Time-dependent Case

Three-dimensional, Time-dependent Case

See also: Differentiation under the integral sign in higher dimensions

A Leibniz integral rule for three dimensions is:

 

  

where:

F ( r, t ) is a vector field at the spatial position r at time t
Σ is a moving surface in three-space bounded by the closed curve ∂Σ
d A is a vector element of the surface Σ
d s is a vector element of the curve ∂Σ
v is the velocity of movement of the region Σ
∇• is the vector divergence
× is the vector cross product
The double integrals are surface integrals over the surface Σ, and the line integral is over the bounding curve ∂Σ.

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