Threedimensional, Timedependent Case
See also: Differentiation under the integral sign in higher dimensionsA 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 threespace 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.
Read more about this topic: Leibniz Integral Rule
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