Question on divergence/gradient theorem

[member 428835]

hey pf!

i had a subtlety about the divergence theroem/gradient theorem. specifically, the following was presented: $$ \iint_s p \vec{dS} = \iiint_v \nabla (p) dv$$

i am familiar with the divergence theorem, but that is for a vector "dotted" with a surface element (flux) related to the divergence (expansion) through the volume. i believe the above is the gradient theorem. can anyone verify the legitimacy of this (and perhaps offer some physical intuition if it holds). i should say p is not a constant.

thanks!

 

[SteamKing]

IDK what 'had a subtlety' means, but here is an article on the divergence theorem:

http://en.wikipedia.org/wiki/Divergence_theorem

 

[arildno]

hey pf!

i had a subtlety about the divergence theroem/gradient theorem. specifically, the following was presented: $$ \iint_s p \vec{dS} = \iiint_v \nabla (p) dv$$

i am familiar with the divergence theorem, but that is for a vector "dotted" with a surface element (flux) related to the divergence (expansion) through the volume. i believe the above is the gradient theorem. can anyone verify the legitimacy of this (and perhaps offer some physical intuition if it holds). i should say p is not a constant.

thanks!

We may write:
\vec{dS}=dS(n_{x}\vec{i}+n_{y}\vec{j}+n_{z}\vec{k})<br /> \int\int_{S}p\vec{dS}=\int\int_{S}pn_{x}dS\vec{i}+\int\int_{S}pn_{y}dS\vec{j}+\int\int_{S}pn_{z}dS\vec{k} (*)<br /> Since the Cartesian unit vector can be extracted from the integrand!<br /> --<br /> But, we have:<br /> pn_{x}dS=(p\vec{i}\cdot\vec{dS})<br /> And the first term in RHS in (*) is then simply, by the divergence theorem:<br /> \int\int_{S}pn_{x}dS\vec{i}=+int\int\int_{V}\frac{\partial{p}}{\partial{x}}\vec{i}dV<br /> <br /> Do the similar thing to the other two terms on RHS in (*), and you get your identity.

 

[member 428835]

thank you arildno! this makes a lot of sense now! youre brilliant!

 

[member 428835]

on another note, arildno, did you just figure this out by thinking about what's going on or did you see it in a book before? either way, thanks again

 

[arildno]

I noticed it a long time ago. I don't remember if it was in a book or on my own inspiration.

 

[vanhees71]

I'd apply Gauß's theorem to the vector field
\vec{V}(\vec{x})=\vec{a} p(\vec{x})
for a constant vector \vec{a}. Then you have
\vec{\nabla} \cdot \vec{V}=\vec{a} \cdot \vec{\nabla} p.
Then Gauß's theorem gives
\int_V \mathrm{d}^3 \vec{x} \vec{a} \cdot \vec{\nabla} p = \vec{a} \cdot \int_V \mathrm{d}^3 \vec{x} \vec{\nabla} p = \int_{\partial V} \mathrm{d}^2 \vec{F} \cdot \vec{a} p = \vec{a} \int_{\partial V} \mathrm{d}^2 \vec{F} p.
Since this is true for all \vec{a} you have proven your vector identity.

 

[arildno]

I'd apply Gauß's theorem to the vector field
\vec{V}(\vec{x})=\vec{a} p(\vec{x})
for a constant vector \vec{a}. Then you have
\vec{\nabla} \cdot \vec{V}=\vec{a} \cdot \vec{\nabla} p.
Then Gauß's theorem gives
\int_V \mathrm{d}^3 \vec{x} \vec{a} \cdot \vec{\nabla} p = \vec{a} \cdot \int_V \mathrm{d}^3 \vec{x} \vec{\nabla} p = \int_{\partial V} \mathrm{d}^2 \vec{F} \cdot \vec{a} p = \vec{a} \int_{\partial V} \mathrm{d}^2 \vec{F} p.
Since this is true for all \vec{a} you have proven your vector identity.

Sure. It's matter of taste, really, which one approach is used.
After all, my method is just as general as the one you present.