Differential form for two variable function where lower and the variable are the same

Question

The attached page is from Steven Shreve's Stochastic calculus.

In the attached page, there is a differential form for a definite integral with two variables: $t$ and $v$'. Integrable variable is $v$ since the integrator is $dv$:

$$ d\left(-\int_t^Tf(t,v)\,dv\right)=f(t,t)\,dt-\int_t^Tdf(t,v)\,dv $$

I am aware how to find differential form for an integral where the limits and the variables are different.

Here the lower limit matches with one of the variables. Hence I struggle to find how the RHS has been derived.

I am happy even if the approach is detailed to me.

Thank you

Differential form two variable integrable function

Answer 1

This can be seen using Leibniz integral rule.

Let $F(t)=-\int_t^T f(t,v)\, dv=\int_T^t f(t,v)\, dv$.

Applying Leibniz integral rule, we get

$$\begin{align} \frac{dF}{dt}&=f(t,t)+\int_T^t\frac{\partial}{\partial t}f(t,v)\,dv\\ &=f(t,t)-\int_t^T\frac{\partial}{\partial t}f(t,v)\, dv \end{align}$$

Note that the book is using the notation that "here and elsewhere in this section, $d$ indicates the differential with respect to the variable $t$" (Section 10.3.2, pg 425).

Converting to differential form, we get the book's result:

$$dF = f(t,t)\,dt-\int_t^Tdf(t,v)\,dv$$