Title: ベクトル解析 (Vector Analysis)
Section: 3次元ユークリッド空間におけるベクトル解析
Subsection: 勾配、回転、発散

=Fをスカラー場とするとき、Fの勾配 gradFをベクトル場                    grad        ⁡        F        =                                            ∂              F                                      ∂                              x                                                    =                  (                                                                      ∂                  F                                                  ∂                                      x                                          1                                                                                            ,                                                            ∂                  F                                                  ∂                                      x                                          2                                                                                            ,                                                            ∂                  F                                                  ∂                                      x                                          3                                                                                                    )                      {\displaystyle \operatorname {grad} F={\partial F \over \partial \mathbf {x} }=\left({\partial F \over \partial x_{1}},{\partial F \over \partial x_{2}},{\partial F \over \partial x_{3}}\right)}  により定義する。さらにベクトル場Xの回転 rot X、発散 div Xをそれぞれベクトル場                    rot        ⁡                  X                =                  (                                                                      ∂                                      X                                          3                                                                                        ∂                                      x                                          2                                                                                            −                                                            ∂                                      X                                          2                                                                                        ∂                                      x                                          3                                                                                            ,                                                            ∂                                      X                                          1                                                                                        ∂                                      x                                          3                                                                                            −                                                            ∂                                      X                                          3                                                                                        ∂                                      x                                          1                                                                                            ,                                                            ∂                                      X                                          2                                                                                        ∂                                      x                                          1                                                                                            −                                                            ∂                                      X                                          1                                                                                        ∂                                      x                                          2                                                                                                    )                      {\displaystyle \operatorname {rot} \mathbf {X} =\left({\frac {\partial X_{3}}{\partial x_{2}}}-{\frac {\partial X_{2}}{\partial x_{3}}},{\frac {\partial X_{1}}{\partial x_{3}}}-{\frac {\partial X_{3}}{\partial x_{1}}},{\frac {\partial X_{2}}{\partial x_{1}}}-{\frac {\partial X_{1}}{\partial x_{2}}}\right)}                      div        ⁡                  X                =                                            ∂                              X                                  1                                                                    ∂                              x                                  1                                                                    +                                            ∂                              X                                  2                                                                    ∂                              x                                  2                                                                    +                                            ∂                              X                                  3                                                                    ∂                              x                                  3                                                                    =        tr        ⁡                              (                                                            ∂                                      X                                          i                                                                                        ∂                                      x                                          j                                                                                            )                                i            j                                {\displaystyle \operatorname {div} \mathbf {X} ={\partial X_{1} \over \partial x_{1}}+{\partial X_{2} \over \partial x_{2}}+{\partial X_{3} \over \partial x_{3}}=\operatorname {tr} \left({\partial X_{i} \over \partial x_{j}}\right)_{ij}}  により定義する。微分演算子ナブラ                    ∇              {\displaystyle \nabla }  を                    ∇        =                  (                                                    ∂                                  ∂                                      x                                          1                                                                                            ,                                          ∂                                  ∂                                      x                                          2                                                                                            ,                                          ∂                                  ∂                                      x                                          3                                                                                                    )                      {\displaystyle \nabla =\left({\partial  \over \partial x_{1}},{\partial  \over \partial x_{2}},{\partial  \over \partial x_{3}}\right)}  と定義すると、勾配、回転、発散は                    grad        ⁡        F        =        ∇        F              {\displaystyle \operatorname {grad} F=\nabla F}                      rot        ⁡                  X                =        ∇        ×                  X                      {\displaystyle \operatorname {rot} \mathbf {X} =\nabla \times \mathbf {X} }                      div        ⁡                  X                =        ∇        ⋅                  X                      {\displaystyle \operatorname {div} \mathbf {X} =\nabla \cdot \mathbf {X} }  と表記できる。