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    Gradient and divergence formulas

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    Derive the gradient formula for a potential in both cylindrical and spherical coordinates. Also, derive the divergence formula of the vector field in both coordinates.

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    Solution Preview

    Please see the attached file for detailed solution.

    (A) Cylindrical coordinates:
    (a) gradient:
    We define the cylindrical coordinates (r, , z) of a point and we have the following relationships
    between rectangular coordinates and cylindrical coordinates:
    We have the following relations
    Hence the unit vector along the r-direction is
    Similarly the unit vector along the  -direction is
    and the unit vector along the ...

    Solution Summary

    The solution shows how to derive the gradient and divergence formulas in both cylindrical and spherical coordinates in detail.