Homework Statement
I just have a little question about Gauss' Law (differential form).
If divE = p/e0 where p is the charge density and e0 is permittivity of free space.
But if we had a sphere with a total net charge of Q, then outside the sphere, the field is E=k/r^2 I think.
Then...
Could someone try and explain with the differential form means? I've only taken p to calculus 2 so I'm not really sure what divergence in the sense of this equation means. Also what is the difference in the two. I mean the integral form looks at an electric field and charge over a region, so...
Homework Statement
Gauss's Law is often given as:
\nabla \cdot \vec{E} = \rho/ \epsilon_0
However E is, in general a function of position, so the equation is really
\nabla \cdot \vec{E}(\vec{r}) = \rho(\vec{r}) /\epsilon_0
correct?
Homework Equations
The Attempt at a Solution
Let M be a smooth manifold. Locally we can choose 1-forms \omega^{1},\omega^{2},...\omega^{n} whish span M^{*}_{q} for each q. Then are there vector fields X_{1}, X_{2}, ...,X_{n} with \omega^{i}(X_{j})=\delta^{i}_{j}? Here \delta^{i}_{j} is Kronecker delta.
By vector fields, I meant vector...
Homework Statement
Let w be the form w= xdydz in R^3. Let S^2 be the unit sphere in R^3.
If we restrict w on S^2, is w exact?
Homework Equations
The Attempt at a Solution
My guess is w is not exact on S^2.
Suppose w is exact on S^2. Then w=da for some 1-form a=fdx+gdy+hdz...
Why does it seem as if the standard differential form of Maxwell's third equation (Faraday's Law) for time varying fields not take into account motional EMF. The differential form simply says that the curl of E is equal to minus the time rate of change of B field. However, there could be a...
Suppose your manifold is just M = R^2 with the standard differential structure (so the atlas is {(R^2, id, R^2)}). Suppose we have a 1-form \omega on M. Then ofcourse \omega = a_1 dx_1 + a_2 dx_2, where the a_i are just c-infinity functions from R^2 to R. Suppose we have a function f on M...