Three-dimensional, Time-dependent 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 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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