What's your favorite thing to do for fun on the weekends? Did you say "change of coordinate systems in double integrals"? THAT'S RIGHT! Time for Jacobians!
We are going to learn how to create the conversion factor for a change of coordinate system in a double integral. For example, you know how when you convert to polars, you throw in r dr dθ? Well, this is like that. EXCEPT MORE.
Essentially, when you change the coordinate system to simplify an integral, you need something to replace dA (which used to be dy dx). And that something must make up for any alteration in the coordinate system you are using. That's where Jacobians come into play!
bullcleo1 explains the purpose of Jacobian transformations and how to calculate a Jacobian for double-integral change of variable problems.
Proof that a change of variables to polar includes r dr dθ:
x(r,θ) = r cos(θ)
∂x/∂r = cos(θ), ∂x/∂θ = -r sin(θ)
y(r,θ) = r sin(θ)
∂y/∂r = sin(θ), ∂y/∂θ = r cos(θ)
Setting up a 2x2 determinant leads to (∂x/∂r • ∂y/∂θ – ∂x/∂θ • ∂y/∂r) = (cos(θ)•r cos(θ) – (-rsin(θ)•sin(θ)) =
r cos2(θ) + r sin2(θ) = r(cos2(θ) + sin2(θ)) = r
We simply add the differentials ∆r∆θ and get our full answer: r dr dθ
This link is a source I found helpful when studying Jacobians and Jacobian determinants. It's not too wordy, so it may help you better understand. The beginning is an example of when and why a transformation of coordinates is appropriate (to simplify a region), and the rest is a review of how we calculate the Jacobian.
Let's assume you are changing something from (x,y) to (u,v).