Line element for Spherical Polar unit vectors using Geometric Algebra.
Posted by peeterjoot on October 9, 2009
The line element for the particle moving on a spherical surface can be calculated by calculating the derivative of the spherical polar unit vector
than taking the magnitude of this vector. We can start either in coordinate form
or, instead do it the fun way, first grouping this into a complex exponential form. Writing , the first factorization is
The unit vector lies in the plane perpendicular to , so we can form the unit bivector and further factor the unit vector terms into a form
This allows the spherical polar unit vector to be expressed in complex exponential form (really a vector-quaternion product)
Now, calcuating the unit vector velocity, we get
The last two lines above factor the vector and the quaternion to the left and to the right in preparation for squaring this to calculate the magnitude.
This last term () has only grade two components, so the scalar part is zero. We are left with the line element
In retrospect, at least once one sees the answer, it seems obvious. Keeping constant the length increment moving in the plane is , and keeping constant, we have . Since these are perpendicular directions we can add the lengths using the Pythagorean theorem.