I Corollaries of the fundamental integral theorems

1. May 19, 2017

Pushoam

Can anyone please tell me significance of these corollaries of fundamental integral theorems?
I can prove these corollaries but I don't understand why do we need to learn it?
Do these corollaries have some physical significance?

(a)$$\iiint_V(\nabla T)d^3 x=\oint_S T d\vec a$$
here S is the surface bounding the volume V .
(b)$$\iiint_V(\nabla \times\vec v)d^3 x= -\oint_S\vec v\times d\vec a$$,where S is the surface bounding the volume V .

(c)$$\iint_S(\nabla T)\times d\vec a =-\oint_P T d\vec l$$ , where P is the boundary of the surface S.
.

2. May 19, 2017

vanhees71

It's sometimes useful in formulating integral laws in fluid dynamics. Be careful with these formulae. They are really safe only when used in Cartesian coordinates!

3. May 19, 2017

o.k.
Thank you.