Question
I think the problem is this: using y bar = y y = y cos + z sin and dy/dy = cos and by inverse rotation matrix y = y cos -z sin anddy/dy = cos so that dy/dy anddy/dy are both cos and one isnot the resiprocal of the other as in solution shown ( I think thisis not a vaild shortcut used in the solution) My del f = ( df /dy cos + df/dz sin )y + (- df/dy sin + df/dz cos) z Which is of course the same del f with coordinates withrespect to new basis y andz Using linear algebra concept of basis vectors... (y compnent of del f , z component of del f) = vector =rotation matrix ( y componenet of del f, z componenetof def f ) = same vector of course. I think the problem is this: using y bar = y y = y cos + z sin and dy/dy = cos and by inverse rotation matrix y = y cos -z sin anddy/dy = cos so that dy/dy anddy/dy are both cos and one isnot the resiprocal of the other as in solution shown ( I think thisis not a vaild shortcut used in the solution) My del f = ( df /dy cos + df/dz sin )y + (- df/dy sin + df/dz cos) z Which is of course the same del f with coordinates withrespect to new basis y andz Using linear algebra concept of basis vectors... (y compnent of del f , z component of del f) = vector =rotation matrix ( y componenet of del f, z componenetof def f ) = same vector of course. Using linear algebra concept of basis vectors... (y compnent of del f , z component of del f) = vector =rotation matrix ( y componenet of del f, z componenetof def f ) = same vector of course.
Explanation / Answer
I believe this is most easily viewed as a change of basisproblem.