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Appendix H




                   Vector Analysis







                   Rectangular Coordinates

                                           f = f(x, y, z),    ~g = g x i + g y j + g z k


                                                        ∂f     ∂f     ∂f
                                                  ∇f =     i +    j +    k
                                                        ∂x     ∂y     ∂z

                                                          ∂g x   ∂g y  ∂g z
                                                 ∇ · ~g =     +      +
                                                          ∂x     ∂y     ∂z

                                                              i   j    k

                                                              ∂  ∂    ∂
                                                   ∇ × ~g =
                                                             ∂x  ∂y  ∂z

                                                             g x g y  g z
                                                                     2
                                                                            2
                                                              2
                                                            ∂ f    ∂ f    ∂ f
                                                       2
                                              ∆f = ∇ f =         +      +
                                                            ∂x 2   ∂y 2    ∂z 2
                   Spherical Coordinates
                                    x = r cos θ sin φ,   y = r sin θ sin φ,   z = r cos φ


                                          f = f(r, θ, φ),    ~g = g r r + g θ θ + g φ φ


                   Divergence Theorem.

                                                ZZ                I
                                                    ∇ · u dx dy =    u · n ds


                   Stoke’s Theorem.

                                                ZZ                  I
                                                    (∇ × u) · ds =     u · dr







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