Suppose you have a parametric equation for a surface, for example:

x = 3*cos(s)+cos(t)*cos(s)

y = 3*sin(s)+cos(t)*sin(s)

z = sin(t)

The partial derivative of this wrt s is:

dsx = -3*sin(s)-cos(t)*sin(s)

dsy = 3*cos(s)+cos(t)*cos(s)

dsz = 0

The partial derivative of this wrt t is:

dtx = -sin(t)*cos(s)

dty = -sin(t)*sin(s)

dtz = cos(t)

Which of these is the tangent in the s direction, and which of these is the tangent in the t direction?

I would have thought the tangent in the s direction is the derivative wrt s above and that the tangent in the t direction is the derivative wrt t, but I seem to remember hearing the opposite.