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General Discussion / Re: Maths thread.
« on: July 23, 2017, 02:40:54 pm »
First of all polar coordinates only really make sense in 2 dimensions, so there are a couple of options for what you mean by polar coordinates in R^3.
So let's stick to the plane. Basically the answer is what you call "B".
First let me take a two-dimensional real vector space V = R^2 with the standard basis whose elements I'll call x and y. Now take another two-dimensional real vector space W = R^2 with the standard basis whose elements I'll call r and theta. Now we define a map f : W --> V by f( ar + b theta) = (a cos b, a sin b).
Now for a point in V we can talk about its polar coordinates, although we are cheating slightly when we do. If we take a point p =(a,b) in V, really it IS the ordered pair (a,b). So when we say its polar coordinates are such and such, we really mean that we are giving the rectangular coordinates of a point q in W with f(q) = p. Since the map f is not injective, this is why polar coordinates are sometimes not well defined.
So let's stick to the plane. Basically the answer is what you call "B".
First let me take a two-dimensional real vector space V = R^2 with the standard basis whose elements I'll call x and y. Now take another two-dimensional real vector space W = R^2 with the standard basis whose elements I'll call r and theta. Now we define a map f : W --> V by f( ar + b theta) = (a cos b, a sin b).
Now for a point in V we can talk about its polar coordinates, although we are cheating slightly when we do. If we take a point p =(a,b) in V, really it IS the ordered pair (a,b). So when we say its polar coordinates are such and such, we really mean that we are giving the rectangular coordinates of a point q in W with f(q) = p. Since the map f is not injective, this is why polar coordinates are sometimes not well defined.