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, <[email protected]> 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 <[email protected]>
>> 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. Is it like this?
>>>
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