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Unformatted text preview: ThreeD Coordinate Systems John E. Gilbert, Heather Van Ligten, and Benni Goetz The graph of a function f ( x, y ) of two variables is a surface in 3space, and familiar single variable methods can be used once a coordinate system has been introduced into 3space. The xyzcoordinate system in 3space is obtained by adding the zaxis perpendicular to the usual xycoordinate system in the xyplane. Each point P in 3space is then determined by a triple ( a, b, c ) as shown to the right. The three coordinate axes intersect at the origin, and each pair of axes determines a coordinate plane . If we drop a perpendicular from P to the the xyplane, we get a point Q with coordinates ( a, b, 0) called the projection of P on the xyplane. In the same way there are projections R (0 , b, c ) and S ( a, , c ) of P on the yzplane and zxplane respectively. x y z P ( a, b, c ) a b c The figures below show the three coordinate planes from two viewpoints : x y z 4 3 2 One Viewpoint: x, y, z > x y z 4 3 21 Another Viewpoint: x, z > , y < Colors help identify the coordinate planes: xyplane { ( x, y, z ) : z = 0 } xzplane { ( x, y, z ) : y = 0 } yzplane { ( x, y, z ) : x = 0 } It will be important to understand planes parallel to the these coordinate planes....
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Sadler
 Multivariable Calculus

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