What is the difference between parametrization and change of variables?

Question

I am embarrassed to ask this, but really need to, in order to clarify my confusion. I am taking multi-variable calculus and I am confused as to the difference between when I should be parametrizing and when I am making a change of variables.

My question is really motivated by a question such as this:

Compute $ \int \int_{E} e^{-4x^2-9y^2}\,dxdy $ where $E$ is the Ellipse $4x^2+9y^2 \le 25$

When I see a question like this, the first thing I think is, ok parametrize the curve, so I simply did:

$x = \frac{5}{2}cos\theta \space$and$ \space y = \frac{5}{3}sin\theta$

and then proceeded to to substitute and do:

$ \int \int_{E} e^{-25}\,rdrd\theta $

getting the wrong result. Actually as I type this, I think I may have answered my own question. Have I got the wrong idea because, I have essentially turned a double integral with two variables into a 1 variable thing? And use an incorrect $r$?

What is a good rule of thumb to keep in mind the difference of when I am making a change of variables and when I am just parametrizing?

Answer 1

In your case, you can change the variables since your surface ($2D$) is represented by $2$ variables (you could call this "parameterization" but "change of variables" is more appropriate). $(x,y) = (\frac52r \cos \theta, \frac53 r\sin \theta)$, $r$ goes from $0$ to $1$ and $\theta$ goes from $0$ to $2\pi$, calculate the Jacobian, and go from there.

You want to parametrize an $n$-dimensional object when it is being represented by more than $n$ variables. This is because you want to work with $n$ variables. For example, take $S$ to be the surface of the unit sphere with $xyz$-cordinates, and you try to calculate $$\iint_S f(x,y,z)dS$$ Then, using the parameterization $(x,y,z) = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$, the integral becomes $$\int_0^{2\pi}\int_0^\pi g(\theta,\phi)\frac{dS}{d\theta d\phi}d\theta d\phi$$ where $\frac{dS}{d\theta d\phi} = \|T_\theta\times T_\phi \|$

Answer 2

paramaterizing is with a single integral

change of variables is a double integral

so for this problem you're meant to use a change of variables, with r and theta. your x and y just need to be multiplied by r. and then chose your boundaries and integrate