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- 713

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I was thinking of the following question: suppose we have a body in two-dimensional space and its density is described by the function [itex]f(x,y)[/itex] (Note! in

*Cartesian*coordinates!). The x-coordinate of the center of mass is given by [tex]C_x= \frac{\int\int xf(x,y)dxdy}{\int\int f(x,y)dxdy}[/tex] where the domain of integration is ℝ

^{2}. The y-coordinate of the center of mass is given by an analogous formula.

Now, suppose we are given the density of this body in

__polar coordinates__, that is:

*f(r,θ)*. Is it possible to obtain directly the centroid [itex](C_r, C_\theta)[/itex] in polar coordinates from

*f(r,θ)*?

Of course I know that by using a polar-to-Cartesian curvilinear transformation

*x=x(r,θ)*,

*y=y(r,θ)*, we can easily obtain the formulas [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex], and then we could just convert C

_{x}and C

_{y}into polar coordinates. However if you think about it, when we calculate [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex] we are essentially calculating the

__Cartesian__

__coordinates of the center of mass__.

Can we "bypass" this step? What's so special here with Cartesian coordinates? Aren't we supposed to be able to calculate the center of mass of a body without resorting to a special coordinate system?