11.7 - 3 is the angle between the positive z-axis and the...

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11.7 Cylindrical and Spherical Coordinates
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The Cylindrical Coordinate System In a cylindrical coordinate system, a point P in space is represented by an ordered triple 1. is a polar representation of the projection P in the xy-plane. 2. z is the directed distance from to P. ( 29 z r , , θ ( 29 , r ( 29 , r
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Figure 11.66
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Conversions: Cylindrical to rectangular Rectangular to cynlidrical z z r y r x = = = θ sin cos z z x y y x r = = + = tan 2 2 2
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Examples: 1) Convert from cylindrical to rectangular coordinates 2) Find an equation in cylindrical coordinates for the rectangular equation 3) Find an equation in rectangular coordinates for the equation in cylindrical coordinates and sketch its graph. - 2 , 4 , 6 π 2 2 2 - + = y x z θ cos 2 = r
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Spherical Coordinates In a spherical coordinate system, a point P in space is represented by an ordered triple 1. is the distance between P and the origin, 0 2. is the same angle used in cylindrical coordinates for r greater than or equal to 0.
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Unformatted text preview: 3. is the angle between the positive z-axis and the line segment from the origin to the point P ) , , ( φ θ ρ ≥ π φ≤ ≤ Figure 11.75 Conversions • Spherical to Rectangular • Rectangular to spherical: φ ρ θ cos sin sin cos sin = = = z y x + + = = + + = 2 2 2 2 2 2 2 arccos tan z y x z x y z y x More Conversions • Spherical to cylindrical (r 0) • Cylindrical to Spherical (r 0) ≥ φ ρ θ cos sin 2 2 2 = = = z r 2 2 2 2 arccos z r z z r + = = + = ≥ More Examples: 1) Convert the point from spherical to rectangular 2) Find an equation in rectangular coordinates and graph. 3) Convert the equation to cylindrical and spherical, ( 29 π , , 9 4 φ ρ sec 2 = z y x = + 2 2...
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11.7 - 3 is the angle between the positive z-axis and the...

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