derivative over nested integrals

Question

I have the following problem. First an example for two-variable functions

Let $A=A(t,\tau)$ and $B=B(t,\tau)$

If I want to compute the following

$\frac{\mathrm{d}}{\mathrm{d}t} \int_0^t \mathrm{d}\tau A(t,\tau) B(t,\tau)$ one can apply the following rule

\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t} \int_0^t \mathrm{d}\tau A(t,\tau) B(t,\tau) = \int_0^t \mathrm{d}\tau \Big( \frac{\partial}{\partial t} A \Big) B + \int_0^t \mathrm{d}\tau A \Big( \frac{\partial}{\partial t} B \Big) + \Big( A(t,\tau) B(t,\tau)\Big)\rvert_{\tau=t} \end{equation}

Now the problem: the functions depend on four time arguments, that is

$A=A(t,\tau,\tau_1,\tau_2)$ and $B=B(t,\tau,\tau_1,\tau_2)$

and I want to compute

\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t} \int_0^t \mathrm{d}\tau \int_\tau^t \mathrm{d}\tau_1 \int_\tau^{\tau_1} \mathrm{d}\tau_2~ A(t,\tau,\tau_1,\tau_2) B(t,\tau,\tau_1,\tau_2) \end{equation}

What would be the rule for this case?

I would really appreciate your help!!!

Stefan

Answer 1

Think of it as

$$\frac{\mathrm{d}}{\mathrm{d}t} \int_0^t \mathrm{d}\tau F(t,\tau,\tau_1)$$

with

$$ F(t,\tau,\tau_1)=\int_\tau^t \mathrm{d}\tau_1 \int_\tau^{\tau_1} \mathrm{d}\tau_2~ A(t,\tau,\tau_1,\tau_2) B(t,\tau,\tau_1,\tau_2) $$

and apply the same principles that you applied in the other case.

Answer 2

As a general principle to avoid mistakes, I would recommend to compute $$ F(t+\delta t)-F(t)$$ and expand it to the first order in $\delta t$.

In your case, for $F(t)=\int_0^t f(t,\tau)\mathrm d\tau$ this gives $$F(t+\delta t)-F(t)=\int_0^t \left[f(t+\delta t,\tau)-f(t,\tau)\right]\mathrm d\tau+\int_t^{t+\delta t}f(t+\delta t,\tau)\mathrm d\tau.$$ From this you deduce immediately that $F'(t)=\int_0^t\partial_tf(t,\tau)\mathrm d\tau+f(t,t)$.