Section 15.8
Change of Variables
The multivariable equivalent of u-substitution.
1
Introduction
In 1D calculus, represents the "stretching factor" of the substitution.
In 2D/3D calculus, this stretching factor is the absolute value of the Jacobian Determinant.
2
The Jacobian
Definition
For a transformation :
Then .
3
Visualizing Transformation
Interactive: Linear Map
4
Worked Examples
Example 1: Polar Coordinates
Verify the Jacobian for .
1. Partial Derivatives:
2. Determinant:
.
Confirmed.
Example 2: Ellipse Area
Find Area of ellipse .
1. Transformation:
Let . Then (Unit Disk D in UV).
2. Jacobian:
.
.
.
3. Integrate:
.
Area of unit disk is .
Result: .
Example 3: Diamond Region
Evaluata where R is bounded by vertices .
1. Describe Lines:
(parallel).
(parallel).
2. Transformation:
Let .
Bounds: , .
3. Jacobian:
Solve for x,y: .
.
.
4. Integrate:
.
.
5
Practice Quiz
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