Page 189 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 189

180                                  VECTORS                               [CHAP. 7



                     JACOBIANS AND CURVINLINEAR COORDINATES

                                               @r  @r
                                      @ðx; y; zÞ       @r
                     7.84.  Prove that                    .
                                              ¼
                                              @u 1 @u 2  @u 3
                                   @ðu 1 ; u 2 ; u 3 Þ
                                                         2
                     7.85.  Express  (a)grad  ;  ðbÞ div A;  ðcÞr   in spherical coordinates.
                                   @     1 @      1  @
                          Ans:                         e 3
                                    @r  e 1 þ  r @   e 2 þ  r sin   @
                               ðaÞ
                                   1 @  2      1  @          1  @A 3
                                                                     where A ¼ A 1 e 1 þ A 2 e 2 þ A 3 e 3
                                   r @r      r sin   @     r sin   @
                               ðbÞ  2  ðr A 1 Þþ   ðsin  A 2 Þþ
                                                                      2
                                   1 @     2 @     1        @     1  @
                                        r              sin
                                   r @r   @r   r sin   @   @   r sin   @
                               ðcÞ  2        þ  2             þ  2  2  2
                     7.86.  The transformation from rectangular to parabolic cylindrical coordinates is defined by the equations
                                   2
                                2
                                                                                            2
                              1
                          x ¼ ðu   v Þ, y ¼ uv, z ¼ z.  (a)Prove that the system is orthogonal.  (b)Find ds and the scale
                              2
                          factors.  (c)Find the Jacobian of the transformation and the volume element.
                                                                         p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
                                    2
                                         2
                                                   2
                                               2
                                                       2
                                                              2
                                                          2
                                            2
                                                                           2
                                                                               2
                                   ds ¼ðu þ v Þ du þðu þ v Þ dv þ dz ;    u þ v ;  h 3 ¼ 1
                          Ans. ðbÞ                                h 1 ¼ h 2 ¼
                                    2
                                            2
                                       2
                                               2
                                   u þ v ;  ðu þ v Þ du dv dz
                               ðcÞ
                                    2
                     7.87.  Write  (a) r   and  (b)div A in parabolic cylindrical coordinates.
                                                      !
                                               2
                                                    2
                                                          2
                                          1    @    @    @
                                    2
                          Ans:  ðaÞ  r   ¼        þ     þ
                                         2
                                        u þ v 2  @u 2  @v 2  @z 2
                                           1     @  p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  @  p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     @A 3
                                                    2   2        2   2
                                         u þ v  @u           @v            @z
                               ðbÞ  div A ¼  2  2  ð u þ v A 1 Þþ  ð u þ v A 2 Þ þ
                     7.88.  Prove that for orthogonal curvilinear coordinates,
                                                           e 1 @   e 2 @   e 3 @
                                                      r  ¼     þ      þ
                                                           h 1 @u 1  h 2 @u 2  h 3 @u 3
                          [Hint: Let r  ¼ a 1 e 1 þ a 2 e 2 þ a 3 e 3 and use the fact that d  ¼r    dr must be the same in both rectangular
                          and the curvilinear coordinates.]
                     7.89.  Give a vector interpretation to the theorem in Problem 6.35 of Chapter 6.
                     MISCELLANEOUS PROBLEMS
                     7.90.  If A is a differentiable function of u and jAðuÞj ¼ 1, prove that dA=du is perpendicular to A.
                     7.91.  Prove formulas 6, 7, and 8 on Page 159.
                                                                                        n
                     7.92.  If   and   are polar coordinates and A; B; n are any constants, prove that U ¼   ðA cos n  þ B sin n Þ
                          satisfies Laplace’s equation.
                                         3
                                2cos   þ 3 sin   cos    2                     2
                     7.93.  If V ¼             , find r V.  Ans.  6 sin   cos  ð4   5 sin  Þ
                                       r 2                             r 4
                     7.94.  Find the most general function of (a)the cylindrical coordinate  ,(b)the spherical coordinate r,(c)the
                          spherical coordinate   which satisfies Laplace’s equation.
                          Ans.(a) A þ B ln  ;  ðbÞ A þ B=r;  ðcÞ A þ B lnðcsc     cot  Þ where A and B are any constants.
                     7.95.  Let T and N denote respectively the unit tangent vector and unit principal normal vector to a space curve
                          r ¼ rðuÞ, where rðuÞ is assumed differentiable. Define a vector B ¼ T   N called the unit binormal vector to
                          the space curve.  Prove that
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