lugita15
May13-10, 02:57 PM
In the midst of this thread (http://physicsforums.com/showthread.php?t=403002), I came upon a rather interesting, though probably elementary, question. Analagous to the fundamental theorem of calculus, is there a formula or theorem concerning the expression \frac{\partial}{\partial t}\int^{t}_{0}F(t, \tau) \partial \tau . In other words, does partial differentiation with respect to one variable commute with partial integration with respect to the other variable? If they don't commute, what is the relation between the two operations?
The fundamental theorem of calculus (well, the first part anyway) came from the non-rigorous intuition that \int^{x+dx}_{x}f(t)dt=f(x)dx. Can a similar intuitive statement be made about \int^{t+dt}_{t}F(t, \tau) \partial \tau? We know it has to be proportional to dt, but what is the constant of proportionality?
The fundamental theorem of calculus (well, the first part anyway) came from the non-rigorous intuition that \int^{x+dx}_{x}f(t)dt=f(x)dx. Can a similar intuitive statement be made about \int^{t+dt}_{t}F(t, \tau) \partial \tau? We know it has to be proportional to dt, but what is the constant of proportionality?