Transitioning Between Rectangular and Polar Coordinates

Example Polar Coordinates

To make this more concrete, consider the case in which you want to switch between
rectangular and polar coordinates. In rectangular coordinates, you use x and y to
locate a point in the plane: in polar coordinates. vow use r. the distance from the
origin, and theta, the angle from the x-axis.
[Graph]
Thus, the change of variables for this equation is x equals r cosine theta and y
equals r sine theta.
x = r cos θ
y = r sin θ
So, in fact, if a function f depends on x and y, then you can plug x and y into r and
theta to get
the function of f: f = f(x, y)
Then you can ask yourself, what is the partial derivative of a function with respect to
a particular variable? You want to take the partial derivative of fwith respect to x
times cosine theta plus the partial derivative of f with respect to y times y sine theta.

δf/δr=df/dx δx/δr+δf/δy δy/δr =f_x cosθ f_y sinθ
In the same way, you can
express derivatives either in terms of x and y or in terms of
r and theta with simple relations between them.