% DESCRIPTION OF THE CURVCOORD WORKSPACE
% Several programs and utility functions using symbolic math are
% provided to analyze vectors and tensors in general curvilinear
% coordinates. The principal programs are: 1) runcoord plots
% coordinate surfaces for typical coordinate systems 2) runmetric
% computes and prints metric tensor properties for a general
% curvilinear coordinate system 3) rundivcurl compares divergence
% and curl of an arbitrary vector computed in both cartesian and
% curvilinear coordinates 4) runconic plots surfaces showing how
% intersections of a cone and a plane produce conic section curves.
%
% Curvilinear coordinates arise when cartesian coordinates (x,y,z)
% are expressed in terms of three auxiliary parameters such as
% (t1,t2,t3). Allowing one of the parameters, say t1, to vary,
% with t2 and t3 held fixed, defines a space curve called the t1
% coordinate line. Similarly, holding t1 constant,while t2 and t3
% vary defines the t1 coordinate surface. Coordinate surfaces
% intersect in coordinate lines which are typically not orthogonal.
% Furthermore,two different sets of base vectors, defined by
% coordinate line tangents and coordinate surface normals, naturally
% occur with curvilinear coordinate formulations. Vector and tensor
% quantities can be conveniently represented by using either set of
% base vectors, with conversion between bases being facilitated by
% use of the metric tensor components. Differentiation of vector
% quantities in curvilinear coordinates requires differentiation of
% both the component magnitudes and the base vectors. This process
% leads to the concept of covariant derivatives and Christoffel
% symbols which are functions of the metric tensor components.
%
% Two different kinds of tools are included in the workspace. The
% first uses numerics to graphically show intersecting coordinate
% surfaces of a general curvilinear coordinate system. Illustrative
% examples such as cylindrical, spherical, toroidal, ellipsoidal,
% and oblate spheroidal coordinates are presented. Although surface
% plots only require numerics, symbolic computation is needed to
% obtain base vectors, metric tensor components, etc. The MATLAB
% Symbolic Math Toolbox, based on Maple software, handles matrices
% of symbolic elements very conveniently. This greatly simplifies
% symbolic operations such as dot and cross products, divergence,
% curl, Christoffel symbols, and covariant derivatives.
%
% A summary of the equations basic to the programs developed here
% can be obtained by the command: help equations. Helpful references
% on the analytical methods used include:
% "Field Theory Handbook" by Parry Moon and Domina E. Spencer
% "Vector Analysis" by Murral Speigel
% "MathWorld" by Eric W. Weisstein (available on the internet)
%
% The principal functions in the workspace are described below.
%
% cart2curv.m convert cartesian components to curvilinear
% cone.m symbolic function for non-orthogonal conical
% coordinates
% cone0.m function to plot non-orthogonal conical
% coordinates
% covardif.m computes covariant derivatives of a vector
% crldivxyz.m divergence and curl of a cartesian vector
% curls.m vector curl in curvilinear coordinates
% curv2cart.m convert curvilinear components to cartesian
% cylin.m symbolic function defining cylindrical
% coordinates
% cylin0.m function used to plot cylindrical coordinates
% diverge.m vector divergence in curvilinear coordinates
% elipcyl.m symbolic function defining elliptic cylinder
% coordinates
% elipcyl0.m function to plot elliptic cylinder coordinates
% edilsod.m symbolic function defining ellipsoidal coordinates
% elipsod0.m function to plot ellipsoidal coordinates
% equations.m the command: help equations lists fundamental
% equations pertaining to curvilinear coordinates
% metric.m computes base vectors, metric tensors, and
% Christoffel symbols for general curvilinear
% coordinates
% notort.m a special non-orthogonal curvilinear coordinate
% system
% oblate.m symbolic function defining oblate spheroidal
% coordinates
% oblate0.m function used to plot oblate spheroidal coordinates
% parab.m symbolic function defining parabolic coordinates
% parabo.m function used to plot parabolic coordinates
% readme.m typing help readme prints this workspace description
% runconic.m illustrates planes intersecting a cone to produce
% the four conic section curves: parabola, hyperbola,
% circle, and ellipse
% runcoord.m interactive program to draw coordinate surfaces for
% several curvilinear coordinate systems
%
% rundivcrl.m program to compare numerical results when divergence
% and curl of a general vector are computed in both
% cartesian and curvilinear coordinates
%
% rundivcrl.tst shows numerical output from progam rundivcrl for
% several coordinate systems
%
% runmetric.m program to compute and print the base vectors, metric
% tensors, and Christoffel symbols for a general
% curvilinear coordinate system.
% runmetric.tst shows extensive symbolic output from program
% runmetric for various coordinate systems. Output for
% some of the coordinate systems is quite lengthy.
% sphr.m symbolic function defining spherical coordinates
% sphr0.m function used to plot spherical coordinates
% testdivcrl.m script to run data cases on program rundivcrl
% testmetric.m script to run data cases on program runmetric
% toroid.m symbolic function defining toroidal coordinates
% toroid0.m function used to plot toroidal coordinates
