A simple example of a curvilinear coordinate system was considered. The coordinate lines are straight, but slanted relative to the x and y axes. The scale factors were computed [Transparancy 2]. By way of comparison, the scale factors for cylindrical and spherical coordinates were also computed.
An expression for arc length along a spacecurve in curvilinear coordinates was given, involving the scale factors [Transparancy 3]. Next, an expression for the gradient in curvilinear coordinates was derived. Knowing the scale factors for spherical coordinates leads immediately to an expression for the gradient in spherical coordinates.
By considering the definition of divergence in terms of net outflux per unit volume, an expression for the divergence in curvilinear coordinates was derived [Transparancies 4 and 5]. Knowing the scale factors for spherical coordinates leads immediately to an expression for the divergence in spherical coordinates.
In section 3.11 of the textbook, an expression for the curl of a vector field in curvilinear coordinates is also derived, starting from the definition of curl as the swirl per unit area.
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