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dRic2
Gold Member
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As far as I can tell the divergence theorem might be one of the most used theorems in physics. I have found it in electrodynamics, fluid mechanics, reactor theory, just to name a few fields... it's literally everywhere. Usually the divergence theorem is used to change a law from integral form to differential (local) form.
Take for example Gauss's law in integral form:
$$\int_{\partial \Omega} \mathbf E \cdot d \mathbf a = \frac {Q_{int}} {\epsilon}$$
and in local form:
$$\nabla \cdot \mathbf E = \frac {\rho} {\epsilon}$$
If I remember correctly the argument goes like this:
1) use divergence theorem
2) write the RHS like a volume integral of some density function (here it is used the charge density but you can do it in different areas of physics and the same argument is used)
3) say something like this: "since this must be true for any control volume the integral can be dropped" (here may lie my doubt)
Now let's stick with Gauss' law to make the example easier. It happens that Gauss' law in integral form is not valid for changing electric fields while the differential form is still valid (locally) even for changing electric fields. This troubles me. Similar arguments can be presented for every other formula I ran into that was derived with this logic.
Since the differential form was derived starting from the integral form there must be something wrong in the proof OR there should be another way to derive the differential form which do not use divergence theorem and doesn't start from an integral law.
Take for example Gauss's law in integral form:
$$\int_{\partial \Omega} \mathbf E \cdot d \mathbf a = \frac {Q_{int}} {\epsilon}$$
and in local form:
$$\nabla \cdot \mathbf E = \frac {\rho} {\epsilon}$$
If I remember correctly the argument goes like this:
1) use divergence theorem
2) write the RHS like a volume integral of some density function (here it is used the charge density but you can do it in different areas of physics and the same argument is used)
3) say something like this: "since this must be true for any control volume the integral can be dropped" (here may lie my doubt)
Now let's stick with Gauss' law to make the example easier. It happens that Gauss' law in integral form is not valid for changing electric fields while the differential form is still valid (locally) even for changing electric fields. This troubles me. Similar arguments can be presented for every other formula I ran into that was derived with this logic.
Since the differential form was derived starting from the integral form there must be something wrong in the proof OR there should be another way to derive the differential form which do not use divergence theorem and doesn't start from an integral law.
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