Skip to main content

Calculus Small Changes, Big Ideas

Bill Shillito

Section 18.1 Coordinate transformations

Introduction goes here.
🔗
  • A transformation takes points \((u,v)\) in one coordinate system and maps them to points \((x,y)\) in another:
    🔗
    \begin{align*} x \amp = g(u,v)\\ y \amp = h(u,v) \end{align*}
    🔗
  • Linear transformations:
    🔗
    \begin{align*} x \amp = au+bv\\ y \amp = cu+dv \end{align*}
    These transformations send grids to parallelogram grids.
    🔗
    🔗
  • Rotation of coordinates:
    \begin{equation*} \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}\cos\phi \amp -\sin\phi\\ \sin\phi \amp \cos\phi\end{bmatrix} \begin{bmatrix}u\\v\end{bmatrix} \end{equation*}
    🔗
    Even simple transformations can dramatically change how curves appear.
    🔗
    🔗
  • Nonlinear transformations can bend and distort regions in more complicated ways.
    🔗
    🔗
  • A useful way to visualize transformations is to track what happens to a grid in the \((u,v)\)-plane as it is mapped into the \((x,y)\)-plane.
    🔗
    🔗
  • Transformations also act on curves. If a curve is described parametrically by \((u(t),v(t))\text{,}\) then its image under the transformation is
    \begin{equation*} (x(t),y(t)) = (g(u(t),v(t)),\,h(u(t),v(t))\bigr). \end{equation*}
    🔗
    In other words, we can understand how a transformation affects a curve by applying it point-by-point.
    🔗
    🔗
  • For example, a straight line in one coordinate system may become a curved path after a nonlinear transformation.
    🔗
    🔗
  • Transformations also act on entire regions. A region in the \((u,v)\)-plane is mapped to a new region in the \((x,y)\)-plane, often with significant distortion.
    🔗
    Understanding how regions change shape will be essential later when we compute integrals using new coordinate systems.
    🔗
    🔗
  • Complex transformations:
    🔗
    🔗
🔗
PrevTopNext