13.7 Spherical Coordinates

# 13.7 Spherical Coordinates - 1 3.7  S p h e r ic a l  C...

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Unformatted text preview: 1 3 .7  S p h e r ic a l  C o o r d in a t e s   S p h e r ic a l   z y x   ρ  ≥  0 ,  ρ  is  t h e  d is t a n c e  o f t h e  p o in t  t o  t h e  o r ig in .  θ  is  t h e  p o s it iv e  a n g l e  b e t w e e n  t h e  p o s it iv e  x  a x is  a n d   the line segment  OP ′ .  0 ≤ φ ≤ π ,  φ  is  t h e  p o s it iv e  a n g l e  b e t w e e n  t h e   p o s it iv e  z  a x is  a n d  t h e  l in e  s e g m e n t  O P .    C o n v e r s i o n   e q u a t i o n s :  z = ρ cosφ r = ρ sin φ x 2 + y 2 + z 2 = ρ2 x = ρ sin φ cos θ y = ρ sin φ sin θ x 2 + y 2 = ρ 2 sin 2 φ   Ex.  Graph  ρ = 3.        3π E x .  G r a p h   θ = 4 .        E x .  G r a p h   φ = 3π 4 .            E x .  G r a p h   φ = π .        E x . C o n v e r t  t h e  e q u a t io n  6 x  =  x 2 +  y 2 t o  s p h e r ic a l   c o o r d in a t e s .                    E x . x 2 +  y 2 +  z 2 =  1       E x .  z  =  √ 2             E x . ( x  –  1 ) 2 +  y 2 +  z 2 =  1                 Ex.  Convert  ρ = 3 to Cartesian coordinates.        E x .  φ =           π 3        ∫∫∫ E f ( ρ,φ,θ ) dV = ∫∫∫ f ( ρ,φ,θ ) ρ 2 sin φ dρdθdφ E   F in d   φ  a n d   ρ  in  t h e  y z  p l a n e   φ  m e a s u r e s  d o w n  fr o m  t h e  p o s it iv e  z ‐ a x is .  ρ  m e a s u r e s  o u t  fr o m  t h e  o r ig in  t h r o u g h  t h e  r e g io n .  θ  is  m e a s u r e d   o n  t h e  x y  p l a n e .    E x .  C h a n g e  t o  s p h e r ic a l  c o o r d in a t e s   3 ∫∫ 9− x 2 −3 − 9 − x 2 ∫ 0 9− x 2 − y 2 z x 2 + y 2 + z 2 dzdydx               E x .  S e t  u p  in t e g r a l s  t o  fin d  t h e  v o l u m e  o f t h e  s o l id   b o u n d e d   b y   x 2   +   y 2   =   z 2   a n d   x 2   +   y 2    =   4 .                               Ex.  Evaluate  ∫∫∫ y E dV  where E is a hollow sphere whose  outside radius is √2 and the inside radius is 1.                                                          Ex.  Set up integrals to find the volume of the solid bounded  above by x2 + y2 + z2 = 4z and below by x2 + y2 = z2.                        Ex.  Set up integrals to find the z coordinate of the center of  mass of the solid in the previous example.              Ex.  Set up integrals to find the volume of the solid inside   x2 + y2 + z2 = 2 and  outside x2 + y2 = 1.                Ex.  Set up integrals to find the volume of the solid bounded  above by z = x2 + y2  and z = 3.                            Do:  1. Find the volume of the sphere with radius a.    2.  Set up an integral to find the volume of just the cone in  the example using x2 + y2 + z2 = 4z and below by x2 + y2 = z2.    3.  Set up integrals to find the volume under the cone and  within the sphere in the same example.    2.            3.          ...
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## This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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