0
$\begingroup$

Could someone explain the concept of "infinitesimal area" to me? For example, in the image below, why does the integral convert to r^3 and not simply r^2? Thanks!

depiction of question

  • 0
    Draw a circular arc between $r$ and $r+dr$ and between $\theta$ and $\theta+d\theta$. Assume it is so small that it is rectangular. Its height I'd $dr$ and its base is $r d\theta$. – 2017-02-02

1 Answers 1

3

enter image description here

What is the area of that box?

When we convert to polar, a small change in $\theta$ wich we will call $d\theta$ corresponds to an arc length of $r\;d\theta$. The area of that box is $r\;dr\;d\theta$

Increasing the level of abstraction:

When we made a $u$ substitution in plane old single variable calcululs we were effectively making a change of the coordinate system. And if we said $u = f(x)$ we also needed to find $du$ that corresponed with that change in the coordinate system. When we move to mulitple integration, we are doing the same thing, only it is more complicated because the variables are interacting.

Rather than calculating a simple $du$ we calculate the "Jacobian"

The Jacobian is the absolute value of the determinant of this matrix.

$\begin{bmatrix} \frac {dx}{du} & \frac {dy}{du}\\\frac {dx}{dv} & \frac {dy}{dv} \end{bmatrix}$

which in this case we are using $r, \theta$ in stead of $u,v$ nonetheless, it is the same idea.

$x = r \cos \theta\\ y = r \sin \theta$

$Det\begin{bmatrix} \cos \theta & \sin\theta \\ -r\sin\theta & r\cos\theta \end{bmatrix} = r$