Partial differentiation of cos (in vector calculus)

1. Dec 5, 2011

Eight

1. The problem statement, all variables and given/known data

So using standard spherical polar co-ordinates, my notes define a sphere as

r(s,t) = aCos(s)Sin(t) i + aSin(s)Sin(t) j + aCos(t) k

and the normal to the surface is given by the cross product of the two partial differentials:

$\partial$r$/\partial$s X $\partial$r$/dt$

So my issue is what is in my notes where:

$\partial$r$/\partial s$= -aSin(s)Sin(t) i + aCos(s)Sin(t) j + aCot(t) k

It is the final part that I don't understand. Why is the partial differential of Cos(t) with respect to s Cot(t)? I would have calculated it as zero.

2. Relevant equations

$\partial$$/$$\partial$s (aCos(t)) = aCot(t) ???

3. The attempt at a solution

All I can think is that it's using a trig identity? As I understand it the k component should only depend upon t, not just from the above equations but in spherical polar co-ordinates in general, at least how I'm picturing it.

Thanks in advance for any help.

2. Dec 6, 2011

Fredrik

Staff Emeritus
I agree with you. That looks completely wrong. It should be 0. And it's such a weird mistake that I kept staring at it for some time, wondering if I had misunderstood something. I guess that's what you're doing too.

3. Dec 6, 2011

Eight

I keep thinking it must be a mistake too, but if it is a mistake then it is deliberate, as it has been repeated in a number of places. And during lectures where we go through the notes it was never picked up on.