How to translate geometric intuitions about vector fields into algebraic equations By considering stereographic projections, I was asked to find a smooth vector field on $S^2$ which vanishes at 1 point, and one that vanishes at 2 points. The intuition, I think, for the vanishing at 1 point is to have all vectors emanating from either the north pole or south pole; for the vanishing at 2 points I think it should be swirling around one of the axes. But I don't know which fields in $\mathbb{R}^2$ will get mapped to these under the push forward of the coordinate maps. Whats the approach to figuring this out?

First, to address the question in your title: I think the only honest answer is that there is no "standard algorithm" for translating intuitions into equations. It takes lots of practice and lots of trial and error. Try to stretch your geometric intuition as far as you can, and then try to write down formulas to prove your intuition correct. The things that hang you up will lead to new insights, which you can feed back into your intuition for the next pass.
Lee Mosher is probably right that stereographic projection is a red herring for the problem of finding a vector field that vanishes at exactly two points -- there are simpler ways to write down such a vector field, such as the one tangent to latitude circles. But to find a vector field that vanishes at exactly one point, stereographic projection can be extremely helpful.  [Hint: think about a coordinate vector field on $\mathbb R^2$.]
