Increments of area dxdy

**rebghb** · October 6th 2010, 09:53 PM

Hello everyone!

I've lately been thinking about this: why isn't $dxdy$ equal to $rdrd\theta$ .

Okay now, $x=r\cos\theta$ and $y=r\sin\theta$ .
So using the derivative rule, we say that $dx=\cos\theta dr-r\sin\theta d\theta$ and that $dy=\sin\theta dr+r\cos\theta d\theta$ ...
But when we carry out $dxdy$ we get something very long...
How is that?

Thanks!

**Prove It** · October 6th 2010, 11:32 PM

Area and volume elements in polar coordinate systems

By the way, $dx\,dy = r\,dr\,d\theta$ IS true.

**HallsofIvy** · October 7th 2010, 04:44 AM

You cannot just multiply differentials like that: If x= u(s, t) and y= v(s, t) then $dxdy= J(x,y;u,v)dsdt$ where J(x,y;u,v) is the Jacobian determinant: $\left|\begin{array}{cc}\frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v}\end{array}\right|$ .

In this case, u and v are r and $\theta$ , respectively, $x= r cos(\theta)$ and $y= r sin(\theta)$ so $J(x, y, r, \theta)= \left|\begin{array}{cc}cos(\theta) & sin(\theta) \\ -r sin(\theta) & r cos(\theta)\end{array}\right|= (cos(\theta)(r cos(\theta))- (sin(\theta))(-r sin(\theta))$ $= r cos^2(\theta)+ r sin^2(\theta)= r$ .

(In more advanced mathematics, differential geometry, we use that to define the "algebra of differentials" in such a way that it is anti-commutative- that is, that ab= -ba. From that $a^2= a(a)= -(a)a= -a^2$ so all squares are 0.

If $x= r cos(\theta)$ then $dx= cos(\theta) dr- r sin(\theta)d\theta$ and if $y= r sin(\theta)$ then $dy= sin(\theta) dr+ r cos(\theta)d\theta$ .

Then $dxdy= (cos(\theta) dr- r sin(\theta)d\theta)(sin(\theta) dr+ r cos(\theta)d\theta)$ . The terms we would get by multiplying dr and dr together or $d\theta$ and $d\theta$ together are 0 leaving $r cos^2(\theta)drd\theta- r sin^2(\theta)d\theta dr$ which, since multiplication is anti-commutative, is the same as $r cos^2(\theta)drd\theta+ r sin^2(\theta)drd\theta= r drd\theta$ .)

**rebghb** · October 7th 2010, 10:21 PM

Dear HallsofIvy,

Thank you for commenting, I am aware of the Jacobian (stretching factor)... But never considered this case.
Regarding differential Geometry, can you recommend me a book (an intro) from calculus III to differential geometry? I would appreciate it...

Best,

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