# Transitioning Between Rectangular and Polar Coordinates

Topics: Calculus

## 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.