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I Just to be sure about the Jacobian matrix and determinant...

  1. Mar 22, 2016 #1
    Ok, I've got these functions to get the x (right), y (up) and z (forward) coordinates to plot with my computer program:

    x = r*Math.cos(a)*Math.sin(o)
    y = r*Math.sin(a)
    z = -r*Math.cos(a)*Math.cos(o)

    It's the equations of a sphere where I've placed the origin (o,a,r) = (0,0,0) of the source (spherical) coordinate system at (x,y,z) = (0,0,-r) in the destination space. From the origin in (a,o,r)-space, Longitude (o) increases in the positive x-direction before heading off into positive z-direction, latitude (a) increases in the positive y-direction and the radius (r) stays fixed.

    To learn about the Jacobian matrix, I first write down the definition:

    [ dx/do dx/da dx/dr ]
    [ dy/do dy/da dy/dr ] =
    [ dz/do dz/da dz/dr ]

    [ r*cos(a)*cos(o) -r*sin(a)*sin(o) cos(a)*sin(o) ]
    [ 0 r*cos(a) sin(a) ]
    [ r*cos(a)*sin(o) r*sin(a)*cos(o) -cos(a)*cos(o) ]

    Now, for a point of derivation on the sphere, drawing the column vectors from that point gives me three basis vectors (not neccessarily orthogonal or with unit length) to the local (o,a,r)-space. Multiplying these vectors respectively with do, da, dr and summing these up gives me a vector in (x,y,z)-space which I draw from the point.

    This process is the equivalent of a change of basis for a vector (do,da,dr) in the local (o,a,r)-space to the global (x,y,z)-space, right?

    It would then make sense if the Jacobian (as in the determinant), used in integration etc. is the absolute value of the _transpose_ Jacobian matrix determinant. It would give me the volume spanned by the parallelpiped of column vectors in the Jacobian matrix (or rows in the transposed matrix) in (x,y,z)-space, corresponding to a 1-sided cube in the (o,a,r)-space.

    Is this correct (many seems to take the determinant of the non-transposed matrix, even with the same definition i've used above)?

    Does it make any sense to draw the row vectors of the Jacobian matrix?

    Interpreting the multiplication of the matrix and a unit vector as a change of basis for a unit vector in local (o,a,r)-space to a "gradient space" in global (x,y,z)-space by projecting the unit vector onto the row vectors (gradients) of the matrix. Since the "gradient space" represents change in (x,y,z)-space divided by change in (a,o,r)-space, the multiplication results in a (x,y,z)-space vector.

    Is there any other intuitive explaination that can be used for the row vectors?
    Is there any use for a determinant of an un-transposed Jacobian matrix?

    Rgds
    Rabbed
     
    Last edited: Mar 22, 2016
  2. jcsd
  3. Mar 23, 2016 #2
    Sorry, I can't seem to edit my post..
    At all places where it says (a,o,r)-space, I mean (o,a,r)-space since the Jacobian matrix columns were created by derivating with respect to o, a and r in that order..
     
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