Mapping a Vector to Various Coordinate Systems
Note it is easy to change to cylindrical or spherical coordinates and then back to cartesian coordinates, even without explicitly defining the components of v. This is accomplished using the MapToBasis command as shown below.
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 end proc], [proc (rho, phi, theta) options operator, arrow; Vector[Column]( 1..3,[ ... ]...](images/Mapping a Vector to Various Coordinate Systems_2.gif){kind=small}
It doesn't look like anything was accomplished, but if we now look at the vector 
we see that Maple has indeed mapped the vector to spherical coordinates.
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^2+vy(x, y, z)^2+vz(x, y, z)^2)^(1/2)], [arctan((vy(x, y, z)^2+vx(x, y, z)^2)^(1/2), vz(x, y, z))], [arctan(vy(x, y, z), vx(x, y, z))]], [](images/Mapping a Vector to Various Coordinate Systems_5.gif){kind=small}
^2+vy(x, y, z)^2+vz(x, y, z)^2)^(1/2)], [arctan((vy(x, y, z)^2+vx(x, y, z)^2)^(1/2), vz(x, y, z))], [arctan(vy(x, y, z), vx(x, y, z))]], [](images/Mapping a Vector to Various Coordinate Systems_6.gif){kind=small}
^2+vy(x, y, z)^2+vz(x, y, z)^2)^(1/2)], [arctan((vy(x, y, z)^2+vx(x, y, z)^2)^(1/2), vz(x, y, z))], [arctan(vy(x, y, z), vx(x, y, z))]], [](images/Mapping a Vector to Various Coordinate Systems_7.gif){kind=small}
For the sake of completion, I show a transformation from spherical coordinates to cylindrical coordinates, followed by a transformation back to cartesian coordinates.
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 end proc], [proc (rho, phi, theta) options operator, arrow; Vector[Column]( 1..3,[ ... ]...](images/Mapping a Vector to Various Coordinate Systems_9.gif){kind=small}
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^2+vx(x, y, z)^2)^(1/2)], [arctan(vy(x, y, z), vx(x, y, z))], [vz(x, y, z)]], [](images/Mapping a Vector to Various Coordinate Systems_11.gif){kind=small}
^2+vx(x, y, z)^2)^(1/2)], [arctan(vy(x, y, z), vx(x, y, z))], [vz(x, y, z)]], [](images/Mapping a Vector to Various Coordinate Systems_12.gif){kind=small}
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![(Typesetting:-mprintslash)([v := proc (x, y, z) options operator, arrow; rtable(1 .. 3, {(1) = vx(x, y, z), (2) = vy(x, y, z), (3) = vz(x, y, z)}, datatype = anything, subtype = Vector[column], storag...](images/Mapping a Vector to Various Coordinate Systems_14.gif){kind=small}
![(Typesetting:-mprintslash)([v := proc (x, y, z) options operator, arrow; rtable(1 .. 3, {(1) = vx(x, y, z), (2) = vy(x, y, z), (3) = vz(x, y, z)}, datatype = anything, subtype = Vector[column], storag...](images/Mapping a Vector to Various Coordinate Systems_15.gif){kind=small}
![(Typesetting:-mprintslash)([v := proc (x, y, z) options operator, arrow; rtable(1 .. 3, {(1) = vx(x, y, z), (2) = vy(x, y, z), (3) = vz(x, y, z)}, datatype = anything, subtype = Vector[column], storag...](images/Mapping a Vector to Various Coordinate Systems_16.gif){kind=small}
![(Typesetting:-mprintslash)([v := proc (x, y, z) options operator, arrow; rtable(1 .. 3, {(1) = vx(x, y, z), (2) = vy(x, y, z), (3) = vz(x, y, z)}, datatype = anything, subtype = Vector[column], storag...](images/Mapping a Vector to Various Coordinate Systems_17.gif){kind=small}
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], [vy(x, y, z)], [vz(x, y, z)]], [](images/Mapping a Vector to Various Coordinate Systems_19.gif){kind=small}
Of course, it is also simple to map a vector that is completely defined in terms of numerical values. This is seen below.
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], [1/4*Pi], [1]], [](images/Mapping a Vector to Various Coordinate Systems_23.gif){kind=small}