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l_notes17 - Lecture XVII Curvilinear Coordinates Change of...

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Lecture XVII Curvilinear Coordinates; Change of Variables As we saw in lecture 16, in E 2 we can use the polar coordinates system. In this system, we have a fixed point O and a fixed ray Ox . The coordinates of a point P are given by r , the distance from P to O , and θ the angle made by Ox to OP . We can OP and Ox , as measured going counterclockwise from change the system of coordiantes from polar to Cartesian through a system of equations: x = r cos θ, y = r sin θ. This is called the defining system . To go from the Cartesian system to the polar one, we use the inverse defining system : r = x 2 + y 2 , θ = arctan y , x for ( x, y ) in the first quadrant. Generally, we can introduce new non-Cartesian coordinates u, v by writing x, y as functions of these new coordiantes: x = g ( u, v ) , y = h ( u, v ) . These equations form the defining system for the new coordiantes. They can be summarized by writting the position vector ˆ j. R ( u, v ) = x ˆ i + yj = g ( u, v ) ˆ i + h ( u, v ) ˆ In working with curvilinear coordinates, it is useful
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