Jacobian: Super Detailed Guide
What is the Jacobian?
The Jacobian is used in calculus when changing variables in multiple integrals. It helps account for how
space stretches or compresses when we switch from one set of variables (like x and y) to another (like
u and v). The Jacobian tells you how much a small area or volume is scaled by the change of variables.
Without it, the integral would give you the wrong value because you're changing the shape and size of
the small regions you’re integrating over. Think of it like a corrector that fixes the scale.
Step-by-Step Instructions: 2D Jacobian
Step 1: Choose new variables
Decide which new variables to use instead of x and y. Common choices are u and v. These new
variables should describe the region more simply or help simplify the integrand. For example, if you see
expressions like x + y or x - y often, try letting u = x + y and v = x - y. Make sure to express x and y in
terms of u and v, not the other way around. This is important because we are rewriting the entire
integral in u and v.
Step 2: Build the Jacobian matrix
Now, compute the partial derivatives of x and y with respect to u and v. This means finding how x
changes as u changes, and how x changes as v changes, and doing the same for y. You put these
partial derivatives into a 2x2 matrix. The matrix looks like this: J = | ∂x/∂u ∂x/∂v | on the first row and |
∂y/∂u ∂y/∂v | on the second row. It’s called the Jacobian matrix. This matrix shows how small squares in
the u-v space map to possibly skewed shapes in x-y space.
Step 3: Compute the determinant
Now find the determinant of the Jacobian matrix. In 2D, the determinant of a 2x2 matrix |a b| / |c d| is
(a*d - b*c). The absolute value of this determinant tells you the factor by which areas are stretched or
shrunk during the transformation. This number, often called J or |J|, will multiply the integrand when you
change the variables in the double integral. It's a crucial step that ensures the new integral gives the
same result as the original one.
Step 4: Rewrite the integral
Once you have |J|, you can now substitute everything into the new variables. Replace x and y in the
function you're integrating with their expressions in terms of u and v. Then, multiply the integrand by |J|.
Also, dx dy becomes du dv. Make sure your limits of integration are changed to the u-v coordinate
system as well. This completes the transformation.