public void SphericalManifold3()
{
Variable r = new Variable("r");
Variable theta = new Variable("theta");
Variable phi = new Variable("phi");
Vector3VariableU coordinates = new Vector3VariableU(r, theta, phi);
Matrix3LL metric = new Matrix3LL(Symbol.One, Functions.Pow(r, 2), Functions.Pow(r * Functions.Sin(theta), 2));
Manifold3 manifold = new Manifold3(metric, coordinates.Del);
Func <int, int, int, Symbol> christoffelSymbols = (i, j, k) =>
{
switch ((i << 16) + (j << 8) + k)
{
case 0x00000101:
return(-r);
case 0x00000202:
return(-r *Functions.Pow(Functions.Sin(theta), 2));
case 0x00010001:
case 0x00020002:
return(1 / r);
case 0x00010202:
return(-Functions.Sin(theta) * Functions.Cos(theta));
case 0x00020102:
return(Functions.Cos(theta) / Functions.Sin(theta));
}
return(Symbol.Zero);
};
ManifoldTests.TestManifold(manifold, christoffelSymbols, (i, j, k, l) => Symbol.Zero, (i, j) => Symbol.Zero, Symbol.Zero);
}
public void SpherePolarManifold2()
{
Variable R = new Variable("R");
Variable theta = new Variable("theta");
Variable phi = new Variable("phi");
Vector2VariableU coordinates = new Vector2VariableU(theta, phi);
Matrix2LL metric = new Matrix2LL(Functions.Pow(R, 2), Functions.Pow(R * Functions.Sin(theta), 2));
Manifold2 manifold = new Manifold2(metric, coordinates.Del);
Func <int, int, int, Symbol> christoffelSymbols = (i, j, k) =>
{
switch ((i << 16) + (j << 8) + k)
{
case 0x00000101:
return(-Functions.Sin(theta) * Functions.Cos(theta));
case 0x00010001:
return(Functions.Cos(theta) / Functions.Sin(theta));
}
return(Symbol.Zero);
};
Func <int, int, int, int, Symbol> riemannTensor = (i, j, k, l) =>
{
switch ((i << 24) + (j << 16) + (k << 8) + l)
{
case 0x00010001:
return(Functions.Pow(Functions.Sin(theta), 2));
case 0x01000001:
return(-Symbol.One);
}
return(Symbol.Zero);
};
Func <int, int, Symbol> ricciTensor = (i, j) =>
{
switch ((i << 8) + j)
{
case 0x00000000:
return(Symbol.One);
case 0x00000101:
return(Functions.Pow(Functions.Sin(theta), 2));
}
return(Symbol.Zero);
};
ManifoldTests.TestManifold(manifold, christoffelSymbols, riemannTensor, ricciTensor, 2 * Functions.Pow(R, -2));
}
public void PlanePolarManifold2()
{
Variable r = new Variable("r");
Variable theta = new Variable("theta");
Vector2VariableU coordinates = new Vector2VariableU(r, theta);
Matrix2LL metric = new Matrix2LL(Symbol.One, Functions.Pow(r, 2));
Manifold2 manifold = new Manifold2(metric, coordinates.Del);
Func <int, int, int, Symbol> christoffelSymbols = (i, j, k) => {
switch ((i << 16) + (j << 8) + k)
{
case 0x00000101:
return(-r);
case 0x00010001:
return(1 / r);
}
return(Symbol.Zero);
};
ManifoldTests.TestManifold(manifold, christoffelSymbols, (i, j, k, l) => Symbol.Zero, (i, j) => Symbol.Zero, Symbol.Zero);
}