##\frac{d\hat{t}}{ds} = \frac{d\hat{t}}{d\theta} \frac{d\theta}{ds}##
##\frac{d\hat{t}}{d\theta} \frac{d\theta}{ds} = \frac{-d\hat{t}}{d\theta} \frac{\nabla n sin(\theta)}{n}##
##\frac{d\hat{t}}{ds} = -\frac{\nabla n sin(\theta)}{n} \frac{d\hat{t}}{d\theta}##
so now I need my expression...
wow, yeah, so after the real product rule, I have
##\frac{d}{ds} (n\hat{t}) = \nabla n cos(\theta) \hat{t} + n \frac{d\hat{t}}{ds}##
so I need an expression that can show how ##\hat{t}## changes with ##s##, but I'm having trouble with that.
You said before you found an expression between...
Okay, so I'm going to reduce ##\frac{d}{ds} (n \hat{t})## down to ##\vec{\nabla}n##
##\frac{d}{ds} (n \hat{t})##
##\frac{dn}{ds} + \frac{d\hat{t}}{ds}##
Using a previous result....
##\frac{-A cos(\theta)}{sin^2(\theta)} \frac{d\theta}{ds} + \frac{d\hat{t}}{ds}##
Using another previous result...
thanks TSny, that tip led me to the right answer!
Now I'm trying to verify the ray equation ##\frac{d}{ds} (nt) = \nabla n## with that result; so far,
##\frac{d\theta}{ds} = \frac{-(dn/dy) sin(\theta)}{n}##
since n is only dependent on the y direction, ##\nabla n = \frac{dn}{dy}##, so...
Homework Statement
A wave travels in a stratified medium whose index of refraction is a function of the coordinate y. Show that the angle ##\theta## between a ray and the y-axis obeys the following law:
## \frac{d\theta}{ds} = \frac{-(dn/dy) sin(\theta)}{n} ## , where the distance s is measured...
But like you said before, when you are walking you are propelling yourself forward; this wouldn't be possible without friction. But since your foot is not slipping while you push (no displacement), the static friction is doing no work.
there will be a normal force out of the stair as you walk on them. However, since your foot is situated, no frictional force is moving through any distance, so no Work is being done there.
the fancy way to write Work is ##W=\int F \cdot d\vec{r} ##
So, my understanding is,
The force is all in the vertical direction due to gravity (there's no mention of any frictional effects on the stairs in the problem), so when the person is walking in the horizontal direction, the angle...
E&M - vector analysis is useful to understand all the derivations
Quantum - all kinds of integrals, linear algebra/matrices, spherical harmonics, Dirac notation
Thermo - so far, a lot of partial derivatives
Here's a graph of apparent (observed) magnitude and redshift. You can see that relative to a matter dominated universe (\Omega_\Lambda=0) a supernova at a given redshift appears dimmer (higher apparent magnitude) in an accelerating universe that contains dark energy (\Omega_M = 0.25...