Hi and thank you for your future help. I'm studying Stokes' theorem and I'm stuck with Stokes extensions. I'm studying Analysis 2, so sometimes Wikipedia, for example, is too complicated.
About Stokes theorem, under certain assumptions:
$$\int_{\Omega} rot(\vec F)\cdot d\vec \sigma = \int_{\delta\Omega^+} \vec F \cdot d\vec\Gamma$$
Now, I read that Stokes' theorem is also the theorem of the gradient, rotor and divergence. Can you please help how those theorems descend from the first one?
More:
$$\int_{\delta\Omega^+} (\vec F \cdot d\vec\Gamma) = \int_{\delta\Omega^+} (F_x, F_y, F_z) \cdot (dx, dy, dz)= \int_{\delta\Omega^+} F_xdx + F_ydy + F_zdz$$
If: $ A = \{x \in [x_1, x_2], y \in [y_1, y_2], z \in [z_1, z_2]\}$
Could I write that:
$$\int_{A} (F_xdx + F_ydy + F_zdz) = \int_{x_1}^{x_2}F_xdx + \int_{y_1}^{y_2}F_ydy + \int_{z_1}^{z_2}F_zdz$$
Or is this an error? Thanks for helping me
About the second question, I thinked about it and I think it's not always possible. Let's take an example:
We have: $\vec F (F_x(x,y,z), F_y(x,y,z), F_z(x,y,z))$
$\int_{\Omega} (Fxdx+Fydy+Fzdz) \neq \int_{x_1}^{x_2}F_x(x,y,z)dx + \int_{y_1}^{y_2}F_y(x,y,z)dy + \int_{z_1}^{z_2}F_z(x,y,z)dz$
Because, usually, $F_{x_i}$ is a function of $(x,y,z)$, and we would integrate only in $dx_i$.