Time derivative of definite integral

In summary: The middle term looks like what you get from the chain rule.In summary, the conversation is discussing the application of the chain rule in definite integrals involving separable functions. The speaker is confused about why the chain rule does not apply when x is a function of t, and the response explains that this is because x is a dummy variable of integration and the chain rule does not apply to it. The speaker also brings up a specific example and the response clarifies the use of partial derivatives and the more general LaGrange's formula in this context.
  • #1
mmwave
647
2
Hi,

Physics books gloss over math. Sometimes it bothers me.

Given a separable function of time and position f = h(x)g(t) then

d / dt of [inte] h(x)g(t)dx = [inte] h(x) dg(t)/dt dx

Where the derivative in the second integral is a partial deriv.

Why the chain rule does not apply is glossed over by the statement that the definite integral of f(x,t) with respect to x results in a new function that is a function of t only. I believe that, but since x can be a function of t I can't see why the chain rule does not apply. Can someone please explain this to me?

To be more specific the book says:

d/dt of [inte] x * psi(x,t)dx = [inte] x * dpsi(x,t)/dt dx
where again the second integral has partial deriv. w.r.t. time t. The '*' above just means times here.
 
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  • #2
x cannot be a function of t because x is a dummy variable of integration.

(However, you do pick up some extra terms that bear resemblance to the chain rule if the limits of integration depend on t)
 
  • #3
Originally posted by Hurkyl
x cannot be a function of t because x is a dummy variable of integration.

(However, you do pick up some extra terms that bear resemblance to the chain rule if the limits of integration depend on t)

Thanks Hurkyl,

The limits are definite ( plus/minus infinity and psi(x,t) goes to zero as x goes to infinity). If x can't be a function of t, then the chain rule says x*dpsi/dt + dx/dt * psi and the second term is zero. Surely the book would have said so, so I must have messed something up in the presentation.

Can you confirm that if f(x,t) is separable into h(x)g(t) then it makes no sense for x to depend on t? It would not be separable in fact if x were a function of t?

Sorry to be dense but if everything I've written is correct the author has made a mountain out of a molehill.
 
  • #4
If your integral is F(t)= [int](from a to b) f(x,t)dx then
dF/dt= [int](from a to b) df/dt dx where "df/dt" is the partial derivative. (In your "separable" example, you don't need partial derivative.)

More generally "LaGrange's Formula" is:
If F(t)= int(from a(t) to b(t)) f(x,t)dx the
dF/dt= int(from a(t) to b(t))df/dtdx+(db/dt)f(t,b(t))-(da/dt)f(t,a(t))

Notice that, here, the limits of integration are also functions of t.
 

1) What is the definition of a time derivative of a definite integral?

The time derivative of a definite integral is a mathematical concept that represents the rate of change of a definite integral with respect to time. In other words, it measures how the value of the integral changes as time passes.

2) How is the time derivative of a definite integral calculated?

The time derivative of a definite integral is calculated by differentiating the integrand with respect to time and then evaluating the resulting expression at the upper and lower limits of the integral.

3) What is the physical interpretation of the time derivative of a definite integral?

The physical interpretation of the time derivative of a definite integral depends on the specific application. In general, it represents the rate of change of a physical quantity that is being accumulated over time, such as velocity, acceleration, or displacement.

4) Can the time derivative of a definite integral be negative?

Yes, the time derivative of a definite integral can be negative. This indicates that the value of the integral is decreasing with time. For example, if the integral represents the distance traveled by an object, a negative time derivative would indicate that the object is moving backwards.

5) What are some real-world applications of the time derivative of a definite integral?

The time derivative of a definite integral has many applications in physics, engineering, and economics. Some examples include calculating the rate of change of volume or mass in a chemical reaction, determining the acceleration of a moving object, and analyzing the change in value of a stock portfolio over time.

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