20e-lecture19

20e-lecture19 - Lecture 19 - Wednesday May 13th...

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[email protected] Key words: Change of variables, Jacobian for implicit transformation, hyperbolic co-ordinates, rotation Key concepts: Know how to deal with implicit transformations 19.1 Classical co-ordinate systems Cylindrical co-ordinates. In cylindrical co-ordinates, a point ( x,y,z ) in R 3 is repre- sented by a triple ( ρ,φ,z ) where ρ denotes the distance to the projection of the point on the xy -plane to the origin, namely ρ = p x 2 + y 2 , φ denotes the angle between the line from the origin to the point ( x,y, 0) and the x -axis, and z is the height of the point ( x,y,z ) above the xy -plane. Therefore with some geometry we obtain x = ρ cos φ y = ρ sin φ z = z as the equations representing the transformation from Cartesian to cylindrical co-ordinates. The Jacobian determinant for this transformation is easily seen to be ρ . We insist there- fore that ρ > 0, and also 0 φ < 2 π . Example 1. Find the volume of the region bounded above by the paraboloid z = x 2 + y 2 , on the sides by the cylinder x 2 + y 2 = 1 and below by the xy -plane. Solution. The volume is R R R D 1 dV where D is the region in question. The region D can be represented easily in cylindrical co-ordinates as D = { ( ρ,φ,z ) : 0 φ < 2 π, 0 < ρ 1 , 0 z ρ 2 } since z x 2 + y 2 represents the region below the paraboloid. Therefore
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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20e-lecture19 - Lecture 19 - Wednesday May 13th...

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