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For a sphere:

x = r*cos(a)*sin(o)

y = r*sin(a)

z = -r*cos(a)*cos(o)

where r is radius, a is latitude and o is longitude, the directional derivative (dx,dy,dz) is the jacobian multiplied by a unit vector (vx,vy,vz), right? So i get:

dx = cos(a)*sin(o)*vx - r*sin(a)*sin(o)*vy + r*cos(a)*cos(o)*vz

dy = sin(a)*vx + r*cos(a)*vy

dz = -cos(a)*cos(o)*vx + r*sin(a)*cos(o)*vy + r*cos(a)*sin(o)*vz

Is this correct?

When I visualize this with a computer program I assume x to the right, y upwards and z into the screen.

I use (vx,vy,vz) := (0, cos(v), sin(v)) for an increasing angle variable v to get a unit length vector where vx = 0 because as I understand (vx,vy,vz) should be a vector in (r,a,o) space and r is not changing.

Then (dx,dy,dz) is pointing out from some point p on the sphere and rotate in a plane (seemingly) tangent to the sphere. When p is close to the poles, the length of (dx,dy,dz) seems to change as v goes from 0 to 2*pi, otherwise it looks like the length is more constant during a revolution around p.

Can someone expain if this is correct behaviour?

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# I Jacobian directional derivative

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