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WHY there is no scalar equation for a line in three dimensions ?

  1. Apr 13, 2007 #1
    WHY there is no scalar equation for a line in three dimensions ???

    :confused:
     
  2. jcsd
  3. Apr 13, 2007 #2
    Perhaps you might like to try rephrasing that in a manner which is understandable.
     
  4. Apr 13, 2007 #3
    How long is a piece of string?
     
  5. Apr 13, 2007 #4
    There is

    ds2 = [itex]\eta_{\alpha\beta}[/itex] dx[itex]\alpha[/itex] dx[itex]\beta[/itex]

    Pete
     
    Last edited: Apr 13, 2007
  6. Apr 18, 2007 #5
    Forty two.
     
  7. Apr 18, 2007 #6
    All pieces of string are of length 42, just in different units.
     
  8. Apr 18, 2007 #7

    HallsofIvy

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    Staff Emeritus
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    Rosicky08, R3 has three dimensions. A curve has one dimension so you have to "lose" 2 dimensions. If you had a single, scalar equation, in three variables, you could solve for any one in terms of the other TWO. That gives you "two degrees of freedom": pick any two coordinates and solve for the third: that one equation allows you to "lose" on dimension. Any single scalar equation describes a two dimensional surface. In order to get one dimension you must have either 2 equations in the three space coordinates so that you could solve for two variables in terms of 1 or 3 equations in the 3 space coordinates and one parameter: 3 equations in 4 variables gives you 4-3= 1 degree of freedom or one dimension.
     
  9. Apr 18, 2007 #8
    Well said. :rofl:
     
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