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Difference between 2 types of differentials?

  1. Apr 20, 2012 #1
    I have a simple question about differentials. I have been taught two ways to find the differential and my questions is in what situations do I use each one?

    simply speaking these are the 2 ways

    1.) just take the partials of each component function and throw them in a matrix

    2.) Let f be the function we want the differential for, where f:S→ℝm. You choose a curve [itex]\alpha[/itex](-ε,ε)→S such that [itex]\alpha[/itex](0)=p and [itex]\alpha[/itex]'(0)= v where p is in S and S is a surface, and v is in TpS.
    Then you compose f with [itex]\alpha[/itex] and take the derivative with respect to t.

    Thank you for your time.
     
  2. jcsd
  3. Apr 20, 2012 #2

    HallsofIvy

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    In 1, it looks like you are referring to the differential of a function from [itex]R^m[/itex] to [itex]R^n[/itex] while in 2, you are specifically referring to a function from [itex]R^1[/itex] to [itex]R^n[/itex].

    Also, 1 is assuming a specific coordinate system while 2 is a general, non-coordinate definition.
     
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