Here is a link to all orthogonal coordinate systems in three dimensions and how to calculate vector calculus operations in each -
https://en.wikipedia.org/wiki/Orthogonal_coordinates On Thu, Mar 23, 2017 at 5:38 PM, Alan Bromborsky <abrombo...@gmail.com> wrote: > If the basis vectors are not orthogonal you need to calculate the > reciprocal basis vectors (equivalent to the inverse of the metric tensor). > If [image: \mathbf{e}_i] are your basis vectors then [image: \mathbf{e}^i]are > the reciprocal basis and > [image: \mathbf{e}_i \cdot \mathbf{e}^j = \delta_{ij}]. Or [image: > \mathbf{e}^i = g^{ij}\mathbf{e}_j] where [image: g^{ij}] is the inverse > of [image: g_{ij}] and the gradient operator is [image: > \frac{\partial}{\partial x^{i}}\mathbf{e}^{i}] where the [image: x^i] are > the coordinates. You need to do the same for orthogonal coordinates but > since the metric tensor > for orthogonal coordinates is diagonal computing the inverse is trivial. > For now I would stick to orthogonal coordinates. > > On Thu, Mar 23, 2017 at 3:50 PM, <szymon.mieszc...@gmail.com> wrote: > >> Do you think, that we should restrict our class to only orthogonal >> curvilinear coordinates (Cartesian, Spherical, Cylindrical) or create class >> as general as possible? >> >> >> W dniu wtorek, 21 marca 2017 23:56:32 UTC+1 użytkownik brombo napisał: >>> >>> What you need to define a coordinate system and vector calculus (div, >>> curl, etc.) is a set of coordinate variables, a corresponding set of basis >>> vectors, the dot products of all the basis vectors in terms of the >>> coordinates (the metric tensor), and the derivatives of the basis vectors >>> as a linear combination of the basis vectors with coefficients that depend >>> only upon the coordinates (derivable from the Christoffel symbols which are >>> derived from the metric tensor). Note that the metric tensor can be >>> derived from the vector manifold function which you can write in >>> rectangular coordinates with coefficients that are functions of the >>> coordinates of the coordinate system you wish to define. Instead of hard >>> coding a particular coordinate system just instantiate a member of the >>> coordinate system class as needed with a given vector manifold function or >>> a given metric tensor. For example if the class is called CoordinateSystem >>> then for a spherical coordinate system you would have - >>> >>> ShericalCooridinates = CoordinateSystem((r*cos(theta) >>> ,r*sin(theta)*cos(phi),r*sin(theta)*sin(phi)),(r,theta,phi)) >>> >>> where (r*cos(theta),r*sin(theta)*cos(phi),r*sin(theta)*sin(phi)) is the >>> vector manifold for spherical coordinates and (r,theta,phi) are the >>> coordinate symbols. The if V is a vector function in terms of the spherical >>> coordinates you could have >>> >>> ShericalCooridinates.div(V) returns the divergence and >>> >>> ShericalCooridinates.curl(V) returns the curl, and if A and B are two >>> vectors in spherical coordinates then >>> >>> ShericalCooridinates.dot(A,B) returns the dot product and >>> >>> ShericalCooridinates.cross(A,B) returns the cross product. >>> >>> On Tue, Mar 21, 2017 at 4:21 PM, Sassi Aissa <sassi.aissa...@gmail.com> >>> wrote: >>> >>>> As I read the description of the Idea, I think I have to implement an >>>> abstract class called: Coordinate System, then create several classes that >>>> represent different types >>>> of Coordinate system and inherent from the so called 'Coordinate >>>> System' class. 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