the Leibniz integral rule for differentiation under the integtal sign states that for an integral of the form $\int_{a(x)}^{b(x)}f(x,t)dt,$ where $-\infty<a(x),b(x)<\infty$ and the integrands are functions dependent on $x$.

$$ \begin{aligned} &\frac{d}{dx}\left(\int_{a(x)}^{b(x)}f(x,t)dt\right) \\ &=f\left(x,b(x)\right)\cdot\frac d{dx}b(x)-f\left(x,a(x)\right)\cdot\frac d{dx}a(x)+\int_{a(x)}^{b(x)}\frac\partial{\partial x}f(x,t)dt \end{aligned} $$

where the partial derivative $\frac{\partial}{\partial x}$ indicates that inside the integral, onlt the variation of $f(x,t)$ with $x$ is considered in taking the derivative{Gottfried Leibniz} .

The following three basic theorems on the interchange of limits are essentially equivalent:

• the interchange of a derivative and an integral
• the change of order of partial derivatives
• the change of order of intefration (Fubini's theorem)

A Leibniz integral rule for a two dimensional surface moving in three dimensional space is

$$ \frac{d}{dt}\iint_{\Sigma(t)}\mathbf{F}(\mathbf{r},t)\cdot d\mathbf{A}=\iint_{\Sigma(t)}\left(\mathbf{F}t(\mathbf{r},t)+\left[\nabla\cdot\mathbf{F}(\mathbf{r},t)\right]\mathbf{v}\right)\cdot d\mathbf{A}-\oint{\partial\Sigma(t)}\left[\mathbf{v}\times\mathbf{F}(\mathbf{r},t)\right]\cdot d\mathbf{s}, $$

where:

• F(r, t) is a vector field at the spatial position r at time t,
• Σ is a surface bounded by the closed curve ∂Σ,
• dA is a vector element of the surface Σ,
• ds 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.