The Change of Variable Formula
Recall the Method of Substitution:
- Change variable:
- Change infinitesimal:
- Change bounds:
, and
We can do the same for multiple integrals.
If we have change of variables
that is one to one (i.e every
where
Suppose we have a coordinate system:
Observe that this is kind of like basis vectors, but
If the region
(note that the arrows are just for visualization, sort of like a parameterization)
Fact: Boundaries are mapped to boundaries
For
and - We have
using
For
and - We have
using
For
and - We have
using
For
and - We have
using
Takeaway: it's not always easy to change the bounds of integration.
But it is straight-forward to change
The Change of Variable Formula and the Jacobian
Given a function that maps
where
which is known as the Jacobian of the transformation.
(see determinant, Partial Derivatives).
Recall that the determinant represents the change in area of a transformation. Also recall from change of basis that changing basis is simply a matrix, where the determinant is the change in area of said matrix, and the change-of-basis matrix can be interpreted as a linear transformation.
So, the Jacobian represents the change in area of the tangent vectors, and we just put it together with the area element
In other words,
Another way to look at this, is when doing Method of Substitution, we differentiated the substitution. The derivative is the single-variable version of the Jacobian.
Recall that a non-zero determinant means a linear transformation is invertible. The same thing applies, where
The transformation to Polar Coordinates:
So
As we've seen before with Double Integrals in Polar Coordinates.
(see Pythagorean identities).
Evaluate
solution
Using our example from before (#^d3d1de),
But we can transform to
Our Jacobian:
So we have:
Evaluate
solution
We don't know how to evaluate integrals over ellipses, but we do know how to evaluate an integral over a circle.
We will map the ellipse onto a unit circle. Let
Our transformation shrinks the domain considerably.
left=-6; right=6;
bottom=-4; top=4;
---
9x^2 + 4y^2 = 36
x^2 + y^2 = 1
The Jacobian is:
This compensates for the shrinking of the domain during integration. Now we have
The domain is circular, and now we can integrate in polar coordinates. Let
Evaluate
using
solution
Adding the two equations gives
Our Jacobian:
What does our new shape look like? We can use the strategy from #^d3d1de to map boundaries to boundaries:
For
so we have a line from the points
For
so we have a line from the points
For
so we have a line from the points
For
so we have a line from the points
which gives us the new shape:
First, we integrate "horizontally" (
Next, we integrate "vertically" (
Now we have