Yesterday I purchased Mathematica v9, for its new vector analysis capabilities. I'm excited.
One of the beauties of vector analysis is that powerful formulas are typically written independent of the coordinate system (be it cartesian, cylindrical, spherical, or other coordinate system). The formulas are simple and elegant.
Take the dot product, for example. The result is independent of the coordinate system employed.
However, it seems to me that Mathematica v9 defines the dot product (and other vector analysis functions) specifically for the cartesian coordinate system. Therefore, to use Mathematica's built-in dot product function, we must first explicitly convert our formulas to the cartesian coordinate system. Then things quickly go from simple elegance to complicated ugly.
I was hoping Mathematica would let me declare vectors that are based upon a spherical coordinate system, and then Mathematica would take care of the awkward ugly stuff when doing dot products, gradients, etc.
Can you steer me toward the best (simple and elegant) Mathematic v9 way to do dot products (gradients, and other vector analysis) within a spherical coordinate system?