# Question:Problem with TransformCoordinates

## Question:Problem with TransformCoordinates

Maple 2015

Hello.
Have a question for use Physics[TransformCoordinates] . For example, in a Cartesian coordinate system is an arbitrary tensor. As you know, in this case the covariant, contravariant and mixed components are the same. We have a coordinate transformation such as transforming our Cartesian coordinate system in the oblique coordinate system. In this simple example, correctly calculated the covariant, contravariant and mixed components (structure: covariant and contravariant), but other mixed components (structure: contravariant and covariant) are calculated is not correct. I checked by hand.

 > restart
 > with(Physics):
 > Setup(mathematicalnotation = true):
 > ds := dx[1]^2+dx[2]^2+dx[3]^2:
 > Setup(coordinates = (X = [x[1], x[2], x[3]]), dimension = 3, metric = ds, spacetimeindices = lowercaselatin, quiet):
 > g_[]:
 > A[a, b] = Matrix(3, 3, {(1, 1) = 2, (1, 2) = 1, (1, 3) = 3, (2, 1) = 2, (2, 2) = 3, (2, 3) = 4, (3, 1) = 1, (3, 2) = 2, (3, 3) = 1})
 (1)
 > Define(%):
 (2)
 > A[]
 (3)
 > A[`~`]
 (4)
 > A[`~a`,b,matrix]
 (5)
 > A[`a`,~b,matrix]
 (6)
 >
 > [y[1] = x[1]-x[2], y[2] = x[2]-x[3], y[3] = x[3]]
 (7)
 > solve((7), {x[1], x[2], x[3]})
 (8)
 > OK
 > B[a,b] = TransformCoordinates((8), A[a, b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))
 (9)
 > OK
 > C[a,b] = TransformCoordinates((8), A[~a,~b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))
 (10)
 > OK
 > D[a,b] = TransformCoordinates((8), A[a,~b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))
 (11)
 > Bug
 > E[a, b] = TransformCoordinates((8), A[~a,b], [y[1], y[2], y[3]], [x[1], x[2], x[3]], simplifier = `@`(`simplify/size`, simplify))
 (12)
 > should be
 > LinearAlgebra:-Transpose(rhs((12)))
 (13)
 >