Find the three partial derivatives \(\frac{\partial}{\partial x}\), \(\frac{\partial}{\partial
y}\), and \(\frac{\partial}{\partial z}\) of \(f(x, y, z) = xyz + x^2 y + y^2 z + z^2
x - xy - yz - xz\).
Find the path integral along the unit circle in the \(xy\) plane of \(F(x, y,
z) = (x + 3) {\hat x} + (y + 2) {\hat y} + (z - 1) {\hat z}\).
Convince yourself that work is the path-integral of force.
Formalize what this means.
Level II
Find the electric field inside and outside a shell of charge.
This is often referred to as the "conducting sphere" case —
"conducting" objects only have charge on their boundary.
Find the electric field inside a plane of finite thickness (not
infinitely thin).
We discussed in class that when charge is uniformly distributed
throughout a sphere, the electric field strength increases linearly
as distance from the center increases (while you are within the
sphere). Could you, and if yes how could you, distribute charge
within a sphere so that the electric field strength inside the
sphere increases quadratically with radius (\(E \sim r^2\)).
Level III
Let the charge density in all space be \(\rho = \rho_0 / r^2 (r + 1)^2\).
Calculate the electric field at an arbitrary point in space.
Calculate the electric field due to a line of charge by integrating
over Coulomb's law. Compare to using Gauss's law.
