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Math Help - Increments of area dxdy

  1. #1
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    Increments of area dxdy

    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!
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  2. #2
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    Area and volume elements in polar coordinate systems

    By the way, dx\,dy = r\,dr\,d\theta IS true.
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  3. #3
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    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.)
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  4. #4
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    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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