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A question about Jacobian when doing coordinates transformation

  1. Apr 7, 2013 #1

    When I do the following transformation:

    X_1=x_1+x_2 \\

    It turns out that the Jacobian ##\partial (X_1,X_2)/\partial (x_1,x_2)## is 1. But we have:

    dx_1dx_1+dx_1dx_2=d(x_1+x_2)dx_2=dX_1dX_2=|\partial (X_1,X_2)/\partial (x_1,x_2)|dx_1dx_2=dx_1dx_2

    So we have ##dx_1dx_1=0##. Is this kind of weird? Why does ##(dx_1)^2## have to be 0?

    Thank you!
    Last edited: Apr 7, 2013
  2. jcsd
  3. Apr 7, 2013 #2


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    It represents the differential area for the parallelogram formed by varying by dx1 and then dx1. Since both sides are the same direction, the area is zero.

    At a higher level the differentials are treated as Grassmann variables (like cross product but yielding tensor instead of vector). Then the Jacobian is built into the algebra.
  4. Apr 8, 2013 #3


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    Strictly speaking "[itex]dx_1dx_1[/itex]" has no meaning! How did it get in that problem?
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