Change in variables

Change in variables

Lessons

Notes:

Transformations

Recall that in Integral Calculus, we can change the variable xx to uu of an integral using u-substitution. In other words, we can change from

f(x)dxf(u)du \int f(x)dx \to \int f(u)du

We would like to do something similar like this with double integrals.

Transformations is about changing from one variable to another. We will first start by transforming regions.


Jacobian of a Transformation

The Jacobian of a transformation x=g(u,v)x=g(u,v) & y=h(u,v)y=h(u,v) is the following:

jacobian of a transformation

It is the determinant of a 2 x 2 matrix.

Change of Variables for a Double Integral

Suppose we want to integrate the function f(x,y)f(x,y) in the region RR under the transformation x=g(u,v)x=g(u,v) & y=h(u,v)y=h(u,v). Then the integral will now become:

Rf(x,y)dA=Sf(g(u,v),h(u,v))d(x,y)d(u,v)dudv\int \int_R f(x,y)dA = \int \int_S f(g(u,v), h(u,v)) \left| \frac{d(x,y)}{d(u,v)}\right| du dv

  • Introduction
    Change in Variables Overview:
    a)
    Transformations
    • Transformation = change one variable to another
    • Similar to u-substitution in integral calculus
    • xyxy-coordinate \to uvuv-coordinate
    • An Example of Change in Variable of Regions

    b)
    Jacobian of a Transformation
    • Definition of Jacobian
    • Determinant of a 2 x 2 matrix
    • Deals with derivatives

    c)
    Change of Variables for a Double Integral
    • All xx's & yy's become uu's & vv's
    • Extra term is absolute value of Jacobian
    • An Example