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What does a jacobian mean?

  1. Feb 18, 2008 #1
    What does a jacobian mean? I know what it IS, as in, if given a set of multivariable functions, I can find out the jacobian, but what does it MEAN?

    And why do we use it to change between coordinate systems (cartesian-> polar =|jacobian of polar|* function in polar coordinates)?
     
  2. jcsd
  3. Feb 18, 2008 #2

    quasar987

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    The short answer is that when you take 2 vectors of R² and compute the determinant of the matrix whose lines or columns are the components of these vectors, you get the area of the parallelogram spanned by these 2 vectors.

    In cartesian coordinates, the area of a little rectangle R is dxdy. It turns out that for a change of coordinates (x,y)<-->(u,v), then the area of the rectangle R in the new coordinates u,v is |J|dudv, where J is the jacobian determinant.

    I recommend the calculus book by Stewart, where this is explained in many drawings with colours and excellent explanations.
     
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