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Spivak's calculus on manifolds

  1. Mar 6, 2009 #1
    I am currently working through spivak's calculus on manifolds (which i love by the way) in one of my class. my question is about his notation for partial derivatives. i completely understand why he uses it and how the classical notation has some ambiguity to it. however, i can't help but thinking of partial derivatives, the chain rule (from R^n to R), and such in terms of the classical notation that i have used for so long. even my professor, though he sometimes uses spivak's notation, slips into the classical notation, especially when he is lecturing without notes.
    my question is what notation do you prefer? i haven't seen spivak's notation anywhere outside this book even though it is such a well known book and the notation has obvious advantages. does anyone else use in their daily lives?
  2. jcsd
  3. Mar 7, 2009 #2


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    Re: Notation

    What is Spivak's notation?
  4. Mar 7, 2009 #3
    Re: Notation

    As long as you know that [tex]\frac{\partial}{\partial x^i}[/tex] depends on all the coordinates [tex]x^1,...,x^n[/tex] and not just on [tex]x^i[/tex], using this notation is okay.
  5. Mar 7, 2009 #4
    Re: Notation

    He uses D1f(x,y,z) to denote the first partial derivative of f.
    So far example you can write D1f(g(x,y),h(x,y)) = D1f(g(x,y),h(x,y))*D1g(x,y) + D2f(g(x,y),h(x,y))*D1h(x,y)
    Whereas in classical notation it would be f(g(x.y),h(x.y)) = f(u.v) where u and v are functions of x any y and this gives us [tex] \frac{\partial{f}}{\partial{x}} = \frac{\partial{f}}{\partial{u}}}\frac{\partial{u}}{\partial{x}} + \frac{\partial{f}}{\partial{v}}}\frac{\partial{v}}{\partial{x}} [/tex]
    but f means two different things on the two sides of the equation

    it is also very helpful if you have something like f(g(x,y) + h(h,y), k(x,y))
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