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    Reading a book on surface integrals I ran into the following formula.

    \displaystyle dS= \frac{dxdy}{\hat{n} \cdot \hat{k}}

    Which seems to be quite handy for evaluating surfaces integrals rather than bringing out partial derivatives and cross products.

    I can't seem to see exactly where this comes from though.

    I mean intuitively it seems to make some sense that ds would depend on dxdy and then the angle between n and k but I don't see exactly why it would look like this in particular.

    Anyone can explain?

    I tried to look online for a sketch as I guess that would clear it up but couldn't see anything.

    thanks.
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    (Original post by poorform)
    Reading a book on surface integrals I ran into the following formula.

    \displaystyle dS= \frac{dxdy}{\hat{n} \cdot \hat{k}}

    Which seems to be quite handy for evaluating surfaces integrals rather than bringing out partial derivatives and cross products.

    I can't seem to see exactly where this comes from though.

    I mean intuitively it seems to make some sense that ds would depend on dxdy and then the angle between n and k but I don't see exactly why it would look like this in particular.

    Anyone can explain?

    I tried to look online for a sketch as I guess that would clear it up but couldn't see anything.

    thanks.
    it is a geometric proof involving the angle between the normal and the k vector.

    it is not long but not easy to explain

    Look for a "glossy type American book on Calculus" although some books use an alternative formula (the one with the square root). They tend to have very good diagrams and their explanations are for "dummies"
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    (Original post by TeeEm)
    it is a geometric proof involving the angle between the normal and the k vector.

    it is not long but not easy to explain

    Look for a "glossy type American book on Calculus" although some books use an alternative formula (the one with the square root). They tend to have very good diagrams and their explanations are for "dummies"
    You referring to the \displaystyle ds= \sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1}~ dxdy one?

    And will do thanks.
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    (Original post by poorform)
    You referring to the \displaystyle ds= \sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2+1} one?

    And will do thanks.
    yes this is the formula (sorry NO LaTeX for me)

    I rarely use it although it is widely used.
    As I said look for a "CALCULUS WITH ANALYTIC GEOMETRY" type book in your University library.

    Otherwise if you have no luck I will dig it up
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    This might help:

    http://onlinelibrary.wiley.com/doi/1...37711.app1/pdf
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    Got it thanks. Managed to dig this up on an e-book after searching.
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    (Original post by poorform)
    Got it thanks. Managed to dig this up on an e-book after searching.
    all good
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    (Original post by poorform)
    Got it thanks. Managed to dig this up on an e-book after searching.
    To my mind this is still just "stating" the important bit ("the projection of dS on the xy plane is dS \cos \gamma"), but if that works for you don't worry about it!
 
 
 
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