The following quote is causing me trouble:
" For instance, suppose we start off at ($0^\circ$N, $0^\circ$W), which is just off the Atlantic coast of equatorial Africa, and rotate $90^\circ$ northwards and then $90^\circ$ eastwards. We end up at ($0^\circ$N, $90^\circ$E), which is in the middle of the Indian Ocean. However, if we start at the same point, and rotate $90^\circ$ eastwards and then $90^\circ$ northwards, we end up at the North pole. "
When I try it I always end up on the north pole. The way I'm doing it is that I take a globe and stare at it on $(0^\circ N,0^\circ E)$ I then rotate it till I stare at the north pole then I rotate earth until I'm still at the north pole and lay on top of the 90º east meridian. This is where I should have ended on the indian ocean according to my quote.
The other problem I have is that if I focus in spherical coordinates, I think of 90ºN as $\phi$=0, if they tell me then to rotate 90ºE, I just think this is so ill-defined.
So my conclusion is that I don't know what rotation even means then.
Can anyone explain this to me?
Thanks.
I believe that what's written doesn't really make much sense. Rotation eastwards clearly means "rotation about the axis of the earth that runs from the N to the S pole, in a direction such that London initially moves generally towards Moscow". But "rotation northwards" would have to mean rotation about some axis whose endpoints are both on the equator. Such a rotation would rotate half the equator northwards, and the other half southwards.
My best guess is that "northwards" rotation in this case means rotation about the axis that goes through (0N, 0E) and the earth's center, and which initially takes the point at (0N, 90E) northwards (so that the entire western half of the equator moves SOUTHwards).
With that interpretation, the two statements given are at least consistent, even if the definition seems silly.
On the good side, if you're willing to ignore the names given to the two rotations, you do at least see that they don't commute, which was probably the point of the description. They don't commute because for your point $P = (0N, 0E)$, we have $$ R_1( R_2(P)) \ne R_2(R_1(P)). $$
Added following comments: To work through the details. $R_1$ is rotation about the NS axis of the earth -- in the direction that the earth turns every day, by 90 degrees. It sends points on the equator to points on the equator, and leaves the poles fixed. The point $(0N, 0E)$ goes to (0N, 90 E).
$R_2$ (as I'm gussing it's intended) is a rotation about a line that goes from $(0N, 0E)$ through the earth's center, and exits at $(0N, 180E)$. Because its a rotation about this line, $R_2$ leaves the points $(0N, 0E)$ and $(0N, 180E)$ unmoved. On the other hand, the point $(0N, 90E)$ moves northward 90 degrees to arrive at $(90N, x)$, where the $x$ indicates that latitude doesn't make sense at the poles.
Note that for the point $Q = (0N, 90W)$, $R_2(Q)$ ends up being the SOUTH pole, so calling it a "northwards rotation" is misleading.
Now let's apply first $R_2$, then $R_1$, to the point $P = (0N, 0E)$. When we apply $R_2$, $P$ does not move (see the description of $R_2$ two paragraphs above!). We then apply $R_1$, and the result is the point $(0N, 90E)$.
By contrast, of we first apply first $R_1$, then $R_2$, to the point $P = (0N, 0E)$, something different happens: When we apply $R_1$, $P$ is moved to $(0N, 90E)$. When we then apply $R_2$, that point is moved to the north pole.
By the way, if you're holding a globe in front of you, let's assume that the polar axis is vertical, and that the line from $(0N, 0E)$ through the center of the globe actually runs east-west in your room. When I talk about applying $R_2$ to a point, I mean "rotate about that east-west-line in your room," regardless of how you've rotated the globe in earlier steps (if any). The same goes for $R_1$: I always mean "rotate about the vertical axis of the room you're sitting in," not "rotate about the north-south axis of the globe".
Perhaps a better way to think of this is that you hold the globe FIXED, and when I say "rotate the point $(0N, 0E)$ eastward", I mean to take that point and move it over the surface of the globe in the way that the eastward globe-rotation would have taken it. You arrive at the point $(0N, 90E)$. If you actually rotated the globe, the point $(0N, 0E)$ always ends up (in globe coordinates) at $(0N, 0E)$. I hope that makes sense to you.