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Can anyone confirm this possible error in 'Vector Calculus' by Matthews?

  1. Aug 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the surface integral of u dot n over S where S is the part of the surface z = x + y2 with z<0 and x>-1, u is the vector field u = (2y+x,-1,0) and n has a negative z component.



    2. Relevant equations

    In the text leading up to the end-of-chapter exercises (where this question arises), n is specified as a unit normal vector.

    3. The attempt at a solution

    The solution claims that ndS = (1,2y,-1)dxdy. But isn't this ignoring the fact that n should be a unit vector? Obviously, the position vector for the surface in parameter form is r = (x,y,x+y2), and n is obtained by taking the partial derivative of r with respect to y, the partial derivative of r with respect to x, and forming the cross product of the two (in that order). But then, shouldn't this resulting vector be divided by the magnitude of the cross product- which, by my calculations happens to be √(2 + 4y2), which is never 1?
     
    Last edited: Aug 14, 2012
  2. jcsd
  3. Aug 14, 2012 #2

    LCKurtz

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    Your surface is parameterized as ##\vec R(x,y) = \langle x, y, x+y^2\rangle##. Read my post #13 in this thread:

    https://www.physicsforums.com/showthread.php?t=611873

    with u = x and v = y and you will see why the book is correct.
     
  4. Aug 15, 2012 #3
    Understood. Thank you, LCKurtz.
     
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