Problem
Use the chain rule to proof
\dot{A}=\partial_t A+v_j\partial_jA_i
Attempt at Solution
\dot{A}=\frac{dA_i}{dt} = \partial_t A_i+\frac{dr_i}{dt}\frac{\partial A_i}{\partial r_i}
Obviously
v_j = \frac{dr_j}{dt}
I'm puzzled where the v_j and partial d_j come in
You can neglect the masses of the satellites I think, since Jupiter is so much more massive than its satellites. Remember that the gradient is just effectivelylog_{10}(a) / log_{10}(p) .
Start by rearranging the kepler equation so you can the gradient part on one side, then take the log...
Shouldn't of said long without a reference if I'm studying physics really :). The paraxial approximation allows the small angle approximation to be used.
paraxial wave equation - Solved
Homework Statement
When a laser beam traveling is traveling in one direction, we can make the paraxial approximation.
Question: Find an expression for the surfaces with constant phase in the beam.
Homework Equations
From a previous part of the...
Oops, I see now I was looking past the obvious. I'll do the calculation in the morning. Is there a reason why they leave out the e^{ikz} from the final equation?
Thanks.
edit: V = V(r,z)e^{ikz} ?
Homework Statement
Homework Equations
http://books.google.co.uk/books?id=4NXHYg70qqIC&pg=PA85&lpg=PA85&dq=paraxial+approximation+wave+equation&source=web&ots=6PbKKzSEz6&sig=bspXdKfxc-IiMV6AmoifMSJTHuk&hl=en&sa=X&oi=book_result&resnum=10&ct=result
The Attempt at a Solution
I...