Change in variables
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Intros
Lessons
- Change in Variables Overview:
- Transformations
- Transformation = change one variable to another
- Similar to u-substitution in integral calculus
- -coordinate -coordinate
- An Example of Change in Variable of Regions
- Jacobian of a Transformation
- Definition of Jacobian
- Determinant of a 2 x 2 matrix
- Deals with derivatives
- Change of Variables for a Double Integral
- All 's & 's become 's & 's
- Extra term is absolute value of Jacobian
- An Example
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Examples
Lessons
- Finding the Transformations
Suppose we have , where is the region bounded by , and . Use the transformation and to determine the new region. - Suppose we have , where is the region bounded by . Use the transformation and to determine the new region.
- Finding the Jacobian
Given that the transformations are and , find the Jacobian. - Given that the transformations are and , find the Jacobian.
- Changing the Variables & Integrating
Evaluate where is the region bounded by using the transformation and . - Evaluate where is the region bounded by using the transformation and .