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13.8 Jacobians

13.8 Jacobians

Author: Luisa Kenausis

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!

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Jacobian Change of Variables

bullcleo1 explains the purpose of Jacobian transformations and how to calculate a Jacobian for double-integral change of variable problems.

Proof of r dr dθ

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

Pauls Online Notes for Jacobians

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.

Integrals with Jacobians: Step-by-Step Instructions

Let's assume you are changing something from (x,y) to (u,v).

  1. Rewrite x and y in terms of u and v: x(u,v) and y(u,v)
  2. Redefine your limits of integration in terms of u and v.
  3. Redefine your integrand in terms of u and v.
  4. Find all 4 partial derivatives with respect to u or v: ∂x/∂u, ∂x/∂v, ∂y/∂u, ∂y/∂v
  5. Set up your 2x2 determinant with your ∂xes on top and your /∂us on the left:
    | ∂x/∂u     ∂x/∂v |
    | ∂y/∂u     ∂y/∂v |
  6. Evaluate the determinant by doing (top left * bottom right) – (top right * bottom left) = ∂x/∂u*∂y/∂v – ∂x/∂v*∂y/∂u
  7. You are left with your Jacobian factor. Now all you need is the ∆u∆v and you can plug it into your new and improved integral.
  8. Integrate away! :D