CYLINDRICAL & SPHERICAL COORDINATES

Transcript

1 CYLINDRICAL & SPHERICAL COORDINATES Here we eamine two of the more popular alternative -dimensional coordinate sstems to the rectangular coordinate sstem. First recall the basis of the Rectangular Coordinate Sstem. Rectangular Coordinates:,, The three orthogonal surfaces intersect at the point: 0, 0, 0 the planes: 0, 0, and 0. and thus specif its location are Page 1 of

2 Clindrical Coordinates: r, θ, The three orthogonal surfaces intersect at the point: r 0, θ 0, 0 the right circular clinder: r r 0, the half plane: θ θ 0, and the plane: 0. and thus specif its location are Page of

3 Spherical Coordinates: ρ, θ, φ The three orthogonal surfaces intersect at the point: ( ρ 0, θ 0, φ 0 ) and thus specif its location are the sphere: ρ ρ 0, the half plane: θ θ 0, and the cone: φ φ 0. Page of

4 Formulas to convert from Clindrical or Spherical Coordinates to Rectangular Coordinates can be readil deduced from the pictures above. Clindrical-to-Rectangular: r cos( θ) r sin( θ) Spherical-to-Rectangular: ρ sin φ cos( θ) ρ sin φ ρ cos( φ) sin( θ) We can use these two sets of conversion formulas to deduce the remaining four conversions as follows. Page of

5 Rectangular-to-Clindrical: ( ) ( ) + r cos θ + r sin θ r r + r cos θ r sin( θ) Rectangular-to-Spherical: + ρ sin φ ( ) + ρ sin φ ( ) ρ cos φ ( cos( θ) ) + ( sin( θ) ) Page 5 of

6 ( ) ( ) + + ρ sin φ + ρ cos φ ρ ρ + + sin( θ) ρsin φ ρsin( φ) cos( θ) ρ cos( φ) cos φ ρ + + cos φ + + Page 6 of

7 Spherical-to-Clindrical: ρ sin φ cos( θ) r cos( θ) r ρ sin( φ) θ θ ρ cos( φ) Clindrical-to-Spherical: ρ ( r cos( θ) ) + ( r sin( θ) ) + ρ r + θ θ + r ρ ( sin φ ) ρ tan φ cos( φ) ( ) tan φ r Page 7 of

8 There are restrictions the we must abide b. Here the are. Clindrical Coordinates Restictions: r 0 0 θ Spherical Coordinates Restictions: ρ 0 0 θ 0 φ The eamples that follow illustrate the relationships between the three coordinate sstems. Page 8 of

9 Eample # 1(a): Convert from Rectangular to Clindrical Coordinates: (,, 1) r + ( ) r + 1 The point lies in the th quadrant of the -plane. θ 1 r, θ, 7,, 1 7 Eample # 1(b): Convert from Rectangular to Clindrical Coordinates: ( 0, 1, 1) r + Page 9 of

10 ( 1) r The -coordinates indicate that: θ 1 ( r, θ, ) 1,, 1 Eample # 1(c): Convert from Rectangular to Clindrical Coordinates: (,, 7) r + ( ) r + 1 The point lies in the nd quadrant of the -plane. Page 10 of

11 θ 7 ( r, θ, ),, 7 Eample # 1(d): Convert from Rectangular to Clindrical Coordinates: (,, ) r + ( ) r + 1 The point lies in the th quadrant of the -plane. θ 7 7 ( r, θ, ),, Page 11 of

12 Eample # (a): Convert from Clindrical to Rectangular Coordinates: 6, 5, 7 r cos θ r sin θ 6 cos sin (,, ),, 7 Eample # (b): Convert from Clindrical to Rectangular Coordinates: 1,, 0 r cos θ r sin θ 1 cos 0 Page 1 of

13 1 sin 1 0 (,, ) 0, 1, 0 Eample # (c): Convert from Clindrical to Rectangular Coordinates:,, 5 r cos θ r sin θ cos 0 sin 5 (,, ) 0,, 5 Page 1 of

14 Eample # (d): Convert from Clindrical to Rectangular Coordinates:,, 1 r cos θ r sin θ cos 0 sin 1 (,, ) 0,, 1 Eample # (a): Convert from Rectangular to Spherical Coordinates: (,, 6). ρ + + cos φ + + Page 1 of

15 ρ The point lies in the 1st quadrant of the -plane. θ cos φ φ φ 6 ( ρ, θ, φ) 8,, 6 Page 15 of E l # (b) C t f R t l t

16 Eample # (b): Convert from Rectangular to Spherical Coordinates: ( 1,, ). ρ + + cos φ ( ) + + ρ The point lies in the th quadrant of the -plane. θ 5 cos φ 0 φ φ ( ρ, θ, φ), 5, Page 16 of E l # C t f R t l t

17 Eample # (c): Convert from Rectangular to Spherical Coordinates: (, 0, 0). ρ + + cos φ + + ( 0) ρ The point lies in the 1st quadrant of the -plane. θ 0 cos φ φ φ ( ρ, θ, φ), 0, Page 17 of E ample # (d) Con ert from Rectang lar to

18 Eample # (d): Convert from Rectangular to Spherical Coordinates: (, 1, ). ρ + + cos φ ( 1) + + ρ The point lies in the 1st quadrant of the -plane. θ 6 cos φ 0 φ φ 6 ( ρ, θ, φ), 6, 6 Page 18 of Eample # : Convert from Spherical to

19 Eample # : Convert from Spherical to Rectangular Coordinates: 1,, ρ sin φ cos θ ρ sin φ ρ cos( φ) sin( θ) ( 1) sin cos ( 1) sin sin ( 1) cos 1 1,,, 6, E l # 5 C t f C li d i l t Page 19 of

20 Eample # 5: Convert from Clindrical to Spherical Coordinates:, 5, 6 ρ r + θ θ tan φ r ( ) ρ + θ 5 6 tan φ 1 φ ( ρ, θ, φ), 5 6, E l # 6 C t f S h i l t Page 0 of

21 Eample # 6: Convert from Spherical to Clindrical Coordinates:,, r ρ sin φ θ θ ρ cos φ sin( ) r 0 θ cos( ) 0 r, θ,,, Eample # 7: Epress the following equation in rectangular coordinates and sketch the graph. r cos( θ) r cos( θ) Page 1 of

22 (, ) Eample # 8: Epress the following equation in rectangular coordinates and sketch the graph. r cos θ ( cos( θ) ) ( sin( θ) ) cos θ Page of

23 , Eample # 9: Epress the following equation in rectangular coordinates and sketch the graph. ρ sec( φ) ρ ρ ρ cos φ (, ) Page of

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