Reynolds Transport derivation?(or 1-form Stokes)

[joob]

Homework Statement

The latex document for these equations wasnt updating correctly, so I've included them as an attachment wherever there is a ... in the text

Ok, I am trying to understand the Reynolds transport theorem, but i don't understand part of it.

Homework Equations

This is the actual equation I am specifically interested in:
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The Attempt at a Solution

Now I know one term would be just the chain rule for f(phi)

...

, but I am not sure how the derivative works for the other term to lower the order to a surface derivative.
From texts I found, I guess the answer should be something like,

...

but I don't really understand how to explicity take the derivative of the integral to lower the order.

 

[mighty2000]

I understand your confusion with the Reynolds transport theorem. It can be a bit tricky to wrap your head around at first, but let me try to explain it to you. The theorem essentially states that the rate of change of a quantity within a control volume is equal to the sum of the local rate of change and the net flux of that quantity across the control volume boundaries.

In terms of the specific equation you mentioned, the first term involving the chain rule is simply the local rate of change of the quantity f(phi) within the control volume. The second term involving the integral is the net flux of f(phi) across the control volume boundaries. This term is usually calculated by integrating over the surface of the control volume, hence the surface derivative.

To explicitly take the derivative of the integral and lower the order, you can use the Leibniz integral rule. This rule states that if the limits of integration of the integral are functions of the variable of differentiation, then the derivative can be written as an integral of the derivative of the integrand over the same limits of integration. In this case, the limits of integration are the control volume boundaries, which are functions of the variable of differentiation (usually time). Therefore, you can rewrite the integral as an integral of the derivative of f(phi) over the control volume boundaries. This will result in the desired surface derivative.

I hope this helps clarify the Reynolds transport theorem for you. Please let me know if you have any further questions or need additional clarification. Best of luck with your studies!