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Homework Help: Projection of surface area elements in vector calculus

  1. Mar 29, 2013 #1
    1. The problem statement, all variables and given/known data
    (i) Find the normal, n, at a general point on the surface S1 given by; x2+y2+z = 1 and z > 0.

    (ii) Use n to relate the size dS of the area element at a point on the surface S1 to its
    projection dxdy in the xy-plane.

    3. The attempt at a solution

    To find n initially I have just done the grad giving 2x,2y,1

    Then with the projection of the surface element I have done that dS=[itex]\hat{n}[/itex]dS and in this case because it is the xy plane it is just in the [itex]\hat{k}[/itex] direction.


    and so I thought dxdy=[itex]\frac{dS}{\sqrt{1+4x+4y}}[/itex]

    I'm really not sure about this so and pointers would be a great help, thanks.

    Also, sorry about the poor LaTeX!
  2. jcsd
  3. Mar 29, 2013 #2


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    I think I'm following you here, but I'm not sure what your two "it"s are referring to.

    This is just the k-component of ##\hat{n}##, right?
    Check your normalization factor here, it's not quite correct.
    Otherwise, I think you're on the right track.
  4. Mar 29, 2013 #3
    Yes sorry, wasn't being very clear, I did mean the k component of n hat there yes. Ok I see the problem with the normalisation factor there, should be 4x2 and 4y2, thank you very much!
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