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, you use r, the distance from the origin, and theta, the angle from the x-axis. 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 f with respect to x times cosine theta plus the partial derivative of f with respect to y times sine theta. ∂𝑓 ∂𝑟 = 𝑑𝑓 ∂𝑥 𝑑𝑥 ∂𝑟 + ∂𝑓 ∂𝑦 ∂𝑦 ∂𝑟 = 𝑓𝑥 𝑐𝑜𝑠 θ 𝑓𝑦 𝑠𝑖𝑛 θ 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.