Where can I use each theorem ?
Could you show me an example?
One major example of the Divergence Theorem is Gauss's Law, which is a flux integral:
$\displaystyle \oint_S \bold{E}.\bold{dA} = \frac{Q_{enclosed}}{\epsilon_0}$
More information on Divergence Theorem here: Divergence theorem - Wikipedia, the free encyclopedia
Stokes I always use to compute flux ? And the divergence theorem ?
example
Determine the flux vector field: $\displaystyle F(x,y,z) = xi +yj + zk$ through the paraboloid $\displaystyle z = x^2+y^2$ below the plane $\displaystyle z = 1$ and vector normal to the surface points to the external side of the paraboloid
I can use Stokes or divergence theorem ?
the flux equal
$\displaystyle flux=\int\int F.n ds$
so how you can use Stokes's theorem for flux is there anti curl because Stoke's
$\displaystyle \oint F.dr=\int\int \nabla\times F .n dS $
since $\displaystyle \nabla\times F$ is the curl of F so if there is an anti for curl you can use Stoke's for flux
but the divergent theorem is
$\displaystyle \int\int F.n dS = \int\int\int div F dv $
$\displaystyle divF =\nabla F$ is the gradient
Stoke's theorem change from surface integral to linear integral but divergence theorem change from surface integral to volume
Green's is a special case from Stoke's
that what I think is correct is there any mistake about I said ...
I think you misunderstand me the flux equal
$\displaystyle \int\int F.n dS \ne \int\int \nabla\times F .n dS $
I need the function that
$\displaystyle F(x,y,z) = \nabla\times G(x,y,z)$ if I can find G I can apply stoke's
$\displaystyle \int\int F.n dS = \int\int \nabla\times G.n.dS = \oint G.dr $
ok show me you should find G such that
$\displaystyle F = \nabla\times G $
you can use Sotke's for flux if they give you a vector like this
$\displaystyle \nabla\times V(x,y,z)$
so for the flux
$\displaystyle \int\int \nabla\times V(x,y,z).n dS = \oint V .dr $
you can't use divergence if the surface is not closed if it is closed the integral above equal zero because
$\displaystyle \int\int \nabla\times V .n .dS = \int\int\int div(\nabla\times V) dv = 0 $
because
$\displaystyle div(\nabla\times V )=0 $
div for the curl equal zero
ok
curl G = F do you know the curl l determinate l
l....... i...... j...... k............ l
l..... d/dx.. d/dy...... d/dz... l = xi + yj + zk
l ...... f...... h........... g..... l
$\displaystyle i(\frac{\partial g}{\partial y} -\frac{\partial h}{\partial z} ) -j(\frac{\partial g}{\partial x} -\frac{\partial f }{\partial z})+k(\frac{\partial h}{\partial x} -\frac{\partial f }{\partial y})=xi+yj+zk
$
$\displaystyle (\frac{\partial g}{\partial y} -\frac{\partial h}{\partial z} ) = x $
$\displaystyle (\frac{\partial g}{\partial x} -\frac{\partial f }{\partial z}) = y$
$\displaystyle (\frac{\partial h}{\partial x} -\frac{\partial f }{\partial y}) = z $
first you should find the derivative with respect to x,y,z then you should find the f ,h, g after you find them but them like this
$\displaystyle F(x,y,z)=f(x,y,z)i + g(x,y,z)j + h(x,y,z)k $
I will try to find them
no
the flux
I will use divergence
$\displaystyle \int\int F.n dS=\int\int\int div(F) dv $
$\displaystyle \int\int\int (\frac{\partial}{\partial x}i+\frac{\partial}{\partial x}j+\frac{\partial}{\partial x}k)(xi+yj+zk) dv $
$\displaystyle \int\int\int \left(\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}\right) dv=\int\int\int (1+1+1) dv=3\int\int\int dv=3V$ V volume of paraboloid
see this link
Paraboloid -- from Wolfram MathWorld