Leibniz's theorem states the time derivative of the integral of \( F(x,t) \) between \( x = a(t) \) and \( x = b(t) \) is
\[
\frac{d}{dt} \int_{x = a(t)}^{x = b(t)} F(x,t)\, dx = \int_{a}^{b} \frac{\partial F}{\partial t} \, dx + \frac{db}{dt} F(b,t) - \frac{da}{dt} F(a,t)
\]
Example: Find
\(
\frac{d}{dt} \int_{x=0}^{x=ct} e^{-x^2} \, dx
\)
This integral cannot be solved in closed form and then differentiated. However, with Leibniz’s rule, the solution is easily found
\(
\because a(t) = 0 , b(t) = ct, F(x,t) = e^{-x^2} \therefore \frac{da}{dt} = 0 , \frac{db}{dt} = c, \frac{\partial F}{\partial t} = 0
\)
\(
\therefore \frac{d}{dt} \int_0^{ct} e^{-x^2} \, dx = 0 + c \cdot e^{-b^2} - 0 = \boxed{c e^{-c^2 t^2}}
\)
The one-dimensional Leibniz theorem can be extended to three dimensions (volume and area integrals) as follows
\[
\frac{d}{dt} \int_{V(t)} F(\vec{x}, t) dV = \int_{V(t)} \frac{\partial F(\vec{x}, t)}{\partial t} dV + \oint_{A(t)} F(\vec{x}, t) \vec{u}_A \cdot d\vec{A}
\]
\[
\underbrace{\frac{d}{dt} \int_{V(t)} F(\vec{x}, t) \, dV}_{\text{Total rate of change of } F}
=
\underbrace{\int_{V(t)} \frac{\partial F(\vec{x}, t)}{\partial t} \, dV}_{\text{Rate of change of } F \text{ due to unsteadiness of } F \text{ itself}}
+
\underbrace{\oint_{A(t)} F(\vec{x}, t) \vec{u}_A \cdot d\vec{A}}_{\text{Rate of change of } F \text{ due to movement of the volume's boundary}}
\]
- \( V(t) \) is some arbitrary volume, which may be changing with time, but not necessarily moving with the fluid
- \( A(t) \) is the surface (area) enclosing volume \( V(t) \); \( A \) is also a function of time since A moves with the volume
- \( d\vec{A} \) is the outward normal vector of a little element of surface area on \( A \)
- \( F(\vec{x}, t) \) is any fluid property (scalar, vector, or tensor of any order)
- F is a function of space and time, independent of what the volume is doing — it is a property of the fluid regardless of what we choose as volume \( V(t) \)
- \( \vec{u}_A \) is the velocity vector defining the motion of surface \( A \)
- This velocity is not necessarily the same as the velocity of the fluid itself, but in general is a function of space and time in an Eulerian frame of reference