Hi all,

Now we are done with Chapter 10, we will move on to analytical geometry. Chapter 10 is mostly introducing a tool, something you can use to simplify your computation, work and therefore life.

If you are given F(x,y)=0, a curve in the normal Cartesian coordinate system, you might want to write it in terms of a new parameter, say t. There are a couple of reasons for this: it is sometimes easier to see the property of the curve (x^2+y^2=1 with x=cos(t) and y=sin(t) clearly tells you the periodicity of the curve), sometime you want to obtain a more dynamic view of the curve (long exposure photo vs. a video), etc. You can, of course, choose to use whichever one in a particular problem.

Every curve y=f(x) always have a trivial parametric equation: x=t and y=f(t). You have a lot of freedom in terms of how to choose your parametric equations x=a(t) and y=b(t). As long as F(a(t),b(t))=0, the parametric equation is legitimate. This is the same idea as substitution in integral calculus, so just like before, make sure to use the chain rule whenever you need to integrate or differentiate.

The idea of polar coordinate system is like parametric equations, just with two parameters r and theta: r measures how far the point is to the origin, and theta is the angle measured in the standard position. The conversion between Cartesian and Polar is always the same: from polar to Cartesian: x=r*cos(theta) and y=r*sin(theta); or from Cartesian to polar: r=sqrt{x^2+y^2}, tan(theta)=y/x. Same idea as the parametric equations, make sure to apply chain rule whenever you use differentiation or integration.

Use either coordinates, as long as it simplifies your work.

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