GM Week 8

GM Week 8

1. Linear Transformations

   • T : R n → R m : T(αu + βv) = αT(u) + βT(v)
     • We also think of T in terms of component functions x = x(u, v) y = y(u, v) where here x(u, v) = au + bv y(u, v) = cu + dv
       • We assume T is invertible and thus we can obtain the component functions of T −1 u = u(x, y) v = v(x, y) although we often do not need such explicit expressions. The transformation is viewed either as a mapping T : R 2 → R 2 or as a change of coordinates in R 2
         • Substitute for x and y in terms of u and v in the integrand f(x, y) = f(x(u, v), y(u, v)) = g(u, v) where we have written g(u, v) to emphasise that after the substitution we have a different function of (u, v) than we had of (x, y).
           • Find the region Γ in (u, v) coordinates corresponding to Ω in (x, y) coordinates: T : Γ → Ω This will determine the limits of integration of u and v.
             • What remains is to express the area element dA in terms of differentials du and dv
               • dA = |J| du dv

2. Extending with Linear Approximations

   • We continue to denote the mapping by T and we assume it maps some region Γ in uv-coordinates to some region of interest Ω in xy-coordinates. We focus on the component functions x = x(u, v) y = y(u, v) which now are not assumed to be linear. Then by linear approximation we have
     • • • dA ≈ ||r1 × r2||

3. Extending Approximations to 3D

   • The generalisation to three dimensions is straightforward. Our transformation T will now have three component functions depending on three variables (u, v, w) x = x(u, v, w) y = y(u, v, w) z = z(u, v, w) T will map some region Γ ⊆ R3 to some region we care about Ω ⊆ R3 .
     • The triple product gives us a determinant of a matrix built out of partial derivatives. The transpose of this is the Jacobian of T for this case and is denoted
       • • Finally, we have

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