Explore BrainMass
Share

Explore BrainMass

    Jacobians and Polar Coordinates

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Please view the attachment for proper formatting of the question.

    Question: The range of the polar coordinates are 0 <= r <= ~, 0 <= theta <= pi, and 0 <= phi <= 2pi and the range of the Cartesian coordinates are - ~ < x,y,z, < ~. In the integrals we must relate the two. This is done by the Jacobian, and the result is,

    dxdydz = (r^2)drsin(theta)d(theta)d(phi)

    Look up the definition of the Jacobian and prove the above result.

    Motivation: Jacobians are the general way of making sure that the volume element in one coordinate system is the same in another. In the case of polar coordinates, the volume element is often called the "solid angle".

    © BrainMass Inc. brainmass.com October 9, 2019, 4:18 pm ad1c9bdddf
    https://brainmass.com/chemistry/general-chemistry/jacobians-polar-coordinates-28717

    Attachments

    Solution Preview

    z = r*cos(theta)

    theta == q, phi == p

    Therefore,

    J =
    | sin(q).cos(p), r.cos(q).cos(p), -r.sin(q).sin(p)|
    | sin(q).sin(p), r.cos(q).sin(p), ...

    Solution Summary

    This solution illustrates which formula is required to solve this integral-based problem and shows all of the calculations required.

    $2.19