Change of Variables

In this activity, we will look at a few familiar Jacobians and their interpretations.

1. Find the Jacobian for the transformation given by $x=3s-1$ and $y=\frac{t}{3}+2$. Write a couple of sentences about why the area of a region will not change with respect to this change of coordinates.

2. Find the Jacobian when changing from rectangular to polar coordinates. Write a couple of sentences to connect to our work on double integr…

Consider the problem of finding the area of the region $D'$ defined by the ellipse $x^2 + \frac{y^2}{4} = 1$. In this activity, we will look at what happens when we use a change of variables so that the pre-image of the domain is a circle.

1. Let $x(s,t) = s$ and $y(s,t) = 2t$. Explain why the pre-image of the original ellipse (which lies in the $xy$ plane) is the circle $s^2 + t^2 = 1$ in the $st$-plane.

2. Recall that the area of the ellipse $D'$ is determined by the double…

Consider the solid $S'$ defined by the inequalities $0 \leq x \leq 2$, $\frac{x}{2} \leq y \leq \frac{x}{2}+1$, and $0 \leq z \leq 6$. Consider the transformation defined by $s = \frac{x}{2}$, $t = \frac{x-2y}{2}$, and $u = \frac{z}{3}$. Let $f(x,y,x) = x-2y+z$.

1. The transformation turns the solid $S'$ in $xyz$-coordinates into a box $S$ in $stu$-coordinates. Apply the transformation to the boundries of the solid $S'$ to find $stu$-coordinate descriptions of the box $S$.

2. Compute and simplify the Jacobian $\frac{\partial(x,y,z)}{\partial(s,t,u)}$.

3. Use the transformation to perform a change of variables…

Let $D'$ be the region in the $xy$-plane that is the parallelogram with vertices $(3,3)$, $(4,5)$, $(5,4)$, and $(6,6)$.

1. Sketch and label the region $D'$ in the $xy$-plane.

2. Consider the integral $\iint_{D'} (x+y) \, dA$. Explain why this integral would be difficult to set up as an iterated integral.

3. Let a change of variables be given by $x = 2u + v$, $y = u + 2v$. Using substitution or elimination, solve this system of equations for $u$ and $v$ in terms of $x$ and $y$.

4. Use your work in (c…

Consider the change of variables which is the transformation from spherical coordinates to rectangular coordinates. Determine the Jacobian of the transformation. How is the result connected to our earlier work with iterated integrals in spherical coordinates?

In this problem, our goal is to find the volume of the ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$.

1. Set up an iterated integral in rectangular coordinates whose value is the volume of the ellipsoid. Do so by using symmetry and taking 8 times the volume of the ellipsoid in the first octant where $x$, $y$, and $z$ are all nonnegative.

2. Explain why it makes sense to use the substitution $x = as$, $y = bt$, and $z = cu$ in order to make the region of integration…