Cylindrical polar co-ordinates

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The discussion focuses on using cylindrical polar coordinates to verify the divergence theorem with a specific vector field, A(r) = x(x-hat) + y(y-hat) + z^2(z-hat). The conversion between Cartesian and cylindrical coordinates is highlighted, particularly the relationships between the unit vectors rho-hat and phi-hat with respect to x and y. The user expresses confusion about the connection between these unit vectors and the Cartesian coordinates. A link to an overview of conversion formulas is provided for further clarification. Understanding these conversions is essential for applying the divergence theorem in cylindrical coordinates.
Nylex
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My notes have an example of verifying the divergence theorem using cylindrical polars.

There's a vector field, A(r) = x(x-hat) + y(y-hat) + z^2(z-hat) and my notes say:

"Note that rho-hat = cos phi(x-hat) + sin phi(y-hat) and phi-hat = -sin phi(x-hat) + cos phi(y-hat) and so

A(r) = rho(rho-hat) + z^2(z-hat)".

Why?? I can't see any obvious connection between phi-hat, rho-hat and x and y :confused:.
 
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