I think I get it based on spherical coordinate transformation,
The first column corresponds to my problem, but I have to add a negative sign because of the way my directions are set-up.
So the only way we have a y-component is if beta != 0 AND alpha != 0, in which case the component along y from the beta part is sin(beta) (because this will be a diagonal vector contributing both to the y component and negatively to the x-component). So I can see the sin(beta) part, but I don't...
On the z-axis it is clearly cos(beta) since that part of the rotation is not influenced by alpha. For the x-axis, I visualize that if alpha=0 then it is -sin(beta) and if alpha != 0 then this is the same as rotating -sin(beta) by cos(alpha). But I cannot figure out the y-axis.
I have been trying to determine an expression for a unit vector in the direction of F for hours now.
I think the expression is supposed to look something kind of like this,
But I don't understand at all how to arrive at this expression.
Any help?
Hm, the only thing I can think of is that if ##k## is optimal and nonzero, then ##g(k+1) + g(k-1) - 2g(k)## is always positive since for optimal ##k## we have ##\Delta g(k) > 0## and ##\Delta g(k-1) < 0##. Is this what you mean?
Yes this is a very meaningful point and I'm glad you brought it up, I should probably have mentioned that the author at the end chose precisely a coordinate system that leads to ##\delta=0## and hence eliminating the occurrence of multiple trigonometric functions, just as you say.
I want to solve ##\frac{du^2}{d\theta ^2}+u=\frac{GM}{h^2}## for ##u(\theta)##, where ##\frac{GM}{h^2}=constant##.
The given equation is a nonhomogeneous second order linear DE. I begin by solving the associated homogeneous DE with constant coefficients:
##\frac{du^2}{d\theta ^2}+u=0##
which...
Fantastic, thank you so much for the reference! I found the book on Google and looked at page 239. Interestingly, I never saw this anywhere in my multivariable calculus course.