Transformations of the plane¶
A transformation of the plane is a one-to-one function that maps one coordinate system to another coordinate system. To represent the transformation $T :\mathbb{R}^2 \to \mathbb{R}^2 :(u,v) \mapsto (x,y)$, we write
$$
T(u,v) = \begin{cases} x = x(u,v), \\ y = y(u,v). \end{cases}
$$
We represent the inverse transformation by
$$
T^{-1}(x,y) = \begin{cases} u = u(x,y), \\ v = v(x,y). \end{cases}
$$
We will sometimes write $(x,y) = T(u,v)$ and $(u,v) = T^{-1}(x,y)$ to represent these relationships.
Video Lecture
1. Example¶
Consider the following transformation.
$$
T(u,v) = \begin{cases} x = u^2 - v^2, \\ y = 2uv. \end{cases}
$$
Find the image of the rectangle $S = [0,1] \times [0,1]$.
Hint: Map each edge of the rectangle individually.
Check your answer
Video solution
The Jacobian of a Transformation¶
Our ultimate goal is to use transformations to change the coordinates of the domain of a function that we wish to integrate. The idea is to change the geometry of the domain of our integral from one that is "complicated" to one that is "simpler."
Suppose $(x,y) = T(u,v)$ is a transformation of the plane, and suppose $z = f(x,y)$ is a continuous function on a domain $D$ in the $xy$-plane. The integral of $f$ over $D$ can be transformed by $T$. It becomes
$$
\int\!\!\!\int_D f(x,y)\, dA_{(x,y)} = \int\!\!\!\int_{T^{-1}(D)} f\big(T(u,v)\big)\, dA_{(u,v)}.
$$
The big question here is "how is $dA_{(x,y)}$ related to $dA_{(u,v)}$?" We explain in this in the next series of videos, but here is the short answer.
$$
dA_{(x,y)} = \left\vert \frac{\partial (x,y)}{\partial (u,v)} \right\vert\, dA_{(u,v)},
$$
where
$$
J(u,v) := \left\vert \frac{\partial (x,y)}{\partial (u,v)} \right\vert = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}.
$$
This function $J(u,v)$ is called the Jacobian, or Jacobian determinant, of the transformation $T$.
Video lecture
Question: Is every transformation of the plane acceptable to use as a change-of-coordinates transformation for an integral?
Video Answer
The answer is NO! This video explains why:
Videos from Khan Academy¶
Here are some very good Khan Academy videos (featuring 3 Blue 1 Brown) describing the Jacobian of a transformation and its effect on area. Please watch them in order and enjoy!
Prerequisites
The Jacobian Matrix
The Jacobian Determinant
In our book the convention is to call the Jacobian matrix by its full name and to call the Jacobian determinant simply the Jacobian.
2. Example¶
Compute the Jacobian (matrix and determinant) of the transformation.
$$
T(u,v) = \begin{cases} x = e^u\sin(u+v) \\[0.5 ex] y = \ln(u^2 + v^2) \end{cases}
$$
Check your answer
The Jacobian matrix is
$$
\widehat{J}(u,v) = \left(\begin{array}{rr}
\cos\left(u + v\right) e^{u} + e^{u} \sin\left(u + v\right) & \cos\left(u + v\right) e^{u} \\
\frac{2 \, u}{u^{2} + v^{2}} & \frac{2 \, v}{u^{2} + v^{2}}
\end{array}\right)
$$
The Jacobian (determinant) is
$$
J(u,v) = -\frac{2 \, u \cos\left(u + v\right) e^{u}}{u^{2} + v^{2}} + \frac{2 \, {\left(\cos\left(u + v\right) e^{u} + e^{u} \sin\left(u + v\right)\right)} v}{u^{2} + v^{2}}
$$
3. Example¶
Compute the integral by making an appropriate change-of-coordinates.
$$
\int\!\!\!\int_D 3x + y^2\, dA,
$$
where $D$ is the triangle in the $xy$-plane with vertices $(0,0)$, $(3,1)$, and $(1,3)$.
Note: This integral does not require a change-of-coordinates to be computed properly, but please use one.
Check your answer
$\dfrac{74}{3}$
Video solution
Discussion¶
Questions? You can ask them below.