Marsden Vector Calculus 6e: Section 1.4 - Exercise 3                                         Page 1 of 4
                Exercise 3
                  (a) The following points are given in cylindrical coordinates; express each in rectangular
                       coordinates and spherical coordinates: (1, 45➦, 1), (2, π/2, −4), (0, 45➦, 10), (3, π/6, 4), (1,
                       π/6, 0), and (2, 3π/4, −2). (Only the first point is solved in the Study Guide.)
                  (b) Change each of the following points from rectangular coordinates to spherical coordinates
                                                                               √              √
                       and to cylindrical coordinates: (2, 1, −2), (0, 3, 4), (  2, 1, 1), (−2  3, −2, 3). (Only the
                       first point is solved in the Study Guide.)
                Solution
                Part (a)
                Cartesian coordinates (x,y,z) and spherical coordinates (ρ,θ,ϕ), with ϕ being the polar angle,
                can be written in terms of cylindrical coordinates (r,θ,z) as
                                              x=rcosθ                         ρ2 = r2 +z2
                                               y = rsinθ                             θ = θ
                                                    z = z                      ρcosϕ=z.
                                                         (r = 1,θ = 45➦,z = 1)
                                     x=1cos45➦                     √        √         !
                                                  
                                     y = 1sin45➦      →       x= 2,y= 2,z=1
                                                                    2        2
                                     z = 1        
                                          p2 2                
                                     ρ =    1 +1              
                                                              
                                                              
                                     θ = 45➦                                √                    
                                                            → ρ= 2,θ=45➦,ϕ=45➦
                                                      1       
                                             −1               
                                     ϕ=cos        √ 2     2   
                                                    1 +1
                                                        (r = 2,θ = π/2,z = −4)
                                            π
                                  x=2cos 
                                              
                                             2
                                  y = 2sin π      → (x=0,y=2,z=−4)
                                            2 
                                              
                                              
                                  z = −4      
                                       p2           2          
                                  ρ =    2 +(−4)               
                                                               
                                                               
                                       π                       
                                                               
                                  θ = 2                                       √         π           
                                                             !      → ρ= 20,θ= ,ϕ≈153➦
                                                                                         2
                                                               
                                          −1         −4        
                                                               
                                  ϕ=cos        p               
                                                   2        2  
                                                  2 +(−4)
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             Marsden Vector Calculus 6e: Section 1.4 - Exercise 3                      Page 2 of 4
                                             (r = 0,θ = 45➦,z = 10)
                               x=0cos45➦
                                         
                               y = 0sin45➦   → (x=0,y=0,z=10)
                                         
                               z = 10    
                                   p2      2       
                               ρ =   0 +10         
                                                   
                                                   
                               θ = 45➦            → (ρ=10,θ=45➦,ϕ=0)
                                                   
                                             10    
                                     −1            
                               ϕ=cos     √ 2     2 
                                           0 +10
                                              (r = 3,θ = π/6,z = 4)
                                      π
                              x=3cos 
                                                      √             !
                                       6
                              y = 3sin π   →     x=3 3,y= 3,z=4
                                      6               2      2
                                        
                                        
                              z = 4     
                                  p2 2            
                              ρ =   3 +4          
                                                  
                                                  
                                  π               
                               θ = 6             → ρ=5,θ=π,ϕ≈36.9➦
                                                                    6
                                                  
                                            4     
                                     −1           
                              ϕ=cos      √ 2   2  
                                          3 +4
                                              (r = 1,θ = π/6,z = 0)
                                        π
                                x=1cos 
                                                      √             !
                                        6
                                y = 1sin π  →     x= 3,y=1,z=0
                                        6              2      2
                                         
                                         
                                z = 0    
                                   p2 2            
                                ρ =  1 +0          
                                                   
                                                   
                                    π              
                                θ = 6               → ρ=1,θ=π,ϕ=π
                                                 
                                                                     6      2
                                                   
                                             0     
                                      −1           
                                ϕ=cos     √ 2    2 
                                            1 +0
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             Marsden Vector Calculus 6e: Section 1.4 - Exercise 3                      Page 3 of 4
                                            (r = 2,θ = 3π/4,z = −2)
                                    3π
                                      
                            x=2cos    
                                      
                                    4 
                                    3π             √      √         
                            y = 2sin   → x=− 2,y= 2,z=−2
                                    4 
                                      
                                      
                            z = −2    
                                p2        2        
                            ρ =  2 +(−2)           
                                                   
                                                   
                                                   
                                                   
                                3π                                             
                            θ = 4                               √      3π     3π
                                                 !     →     ρ =  8,θ =   , ϕ =
                                                                       4       4
                                                   
                                          −2       
                                  −1               
                            ϕ=cos     p            
                                                   
                                         2      2  
                                        2 +(−2)
             Part (b)
             Cylindrical coordinates (r,θ,z) and spherical coordinates (ρ,θ,ϕ), with ϕ being the polar angle,
             can be written in terms of Cartesian coordinates (x,y,z) as
                                    2    2   2               2    2   2   2
                                   r =x +y                  ρ =x +y +z
                                     tanθ = y                      tanθ = y
                                            x                             x
                                         z = z                   ρcosϕ=z.
                                              (x = 2,y = 1,z = −2)
                             p2 2 
                          r =  2 +1 
                                       
                                       
                                    
                                −1  1             √                 
                          θ = tan   2  →        r =  5,θ ≈ 26.6➦,z = −2
                                       
                                       
                          z = −2       
                              p2 2          2        
                          ρ =   2 +1 +(−2)           
                                                     
                                                   
                                                     
                                    1                
                                −1                   
                          θ = tan   2                 → (ρ=3,θ≈26.6➦,ϕ≈132➦)
                                                    !
                                                     
                                                     
                                                     
                                −1         −2        
                                                     
                          ϕ=cos     p                
                                       2   2       2 
                                      2 +1 +(−2)
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             Marsden Vector Calculus 6e: Section 1.4 - Exercise 3                      Page 4 of 4
                                              (x = 0,y = 3,z = 4)
                                p2 2 
                            r =   0 +3 
                                         
                                      
                                   −1  3                  π      
                            θ = tan    0  →       r = 3,θ = 2,z = 4
                                         
                                         
                            z = 4        
                                p2 2 2              
                            ρ =   0 +3 +4           
                                                    
                                                  
                                                    
                                       3            
                                   −1                                π         
                            θ = tan    0                → ρ=5,θ= ,ϕ≈36.9➦
                                                                    2
                                                    
                                                    
                                            4       
                                   −1               
                            ϕ=cos      √ 2   2    2 
                                        0 +3 +4
                                              (x = √2,y = 1,z = 1)
                              q√         
                                    2   2
                          r =   ( 2) +1 
                                         
                                         
                                 −1 1             √                
                          θ = tan    √  → r= 3,θ≈35.3➦,z=1
                                      2 
                                         
                                         
                          z = 1          
                              q√                      
                                    2   2   2         
                          ρ =   ( 2) +1 +1            
                                                      
                                                    
                                                      
                                                      
                                      1               
                                 −1                   
                          θ = tan    √                                          π
                                    2                   → ρ=2,θ≈35.3➦,ϕ=
                                                                                 3
                                                      
                                                      
                                            1         
                                 −1                   
                          ϕ=cos q                  
                                                      
                                        √             
                                       ( 2)2 +12 +12  
                                                   √
                          q                 (x = −2 3,y = −2,z = 3)
                                √          
                                   2      2
                       r =  (−2 3) +(−2) 
                                           
                                                                    
                                 −1 1   →         r = 4,θ = 7π,z = 3
                       θ = π +tan    √                        6
                                      3    
                                           
                                           
                       z = 3               
                          q √                           
                                   2      2   2         
                       ρ =  (−2 3) +(−2) +3             
                                                        
                                                      
                                                        
                                                        
                                      1                 
                                 −1                                                 
                       θ = π +tan    √                                    7π
                                     3                    →     ρ = 5,θ =   , ϕ ≈ 53.1➦
                                                                           6
                                                        
                                                        
                                           3            
                             −1                         
                                                        
                       ϕ=cos q                       
                                                        
                                       √                
                                          2      2   2  
                                   (−2 3) +(−2) +3
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