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ベクトル解析公式集

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物理・数学・プログラムのページについて

概要

ベクトル公式です。ベクトルの演習は公式を覚えれば簡単ですよ。

基本的な公式

    \boldsymbol{ a } \times (\boldsymbol{ b } \times \boldsymbol{ c }) = (\boldsymbol{ a } \cdot \boldsymbol{ c })\boldsymbol{ b } - (\boldsymbol{ a } \cdot \boldsymbol{ b })\boldsymbol{ c }
    (\boldsymbol{ a } \times \boldsymbol{ b }) \times \boldsymbol{ c } = (\boldsymbol{ a } \cdot \boldsymbol{ c })\boldsymbol{ b } - (\boldsymbol{ b } \cdot \boldsymbol{ c })\boldsymbol{ a }
    (\boldsymbol{ a },\boldsymbol{ b },\boldsymbol{ c }) = \boldsymbol{  } \cdot (\boldsymbol{ b } \times \boldsymbol{ c }) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 &b_3 \\ c_1 & c_2 &c_3 \end{vmatrix}
    \nabla (fg) = (\nabla f)g + f(\nabla g)
    \nabla (\frac{ f }{ g }) = \frac{ f (\nabla f)g - f(\nabla g)}{ g^2 }
    \nabla (\varphi (f)) = \varphi '(f) \nabla \varphi (f)
    \nabla (f \boldsymbol{ F }) = (\nabla f) \cdot \boldsymbol{ F } + f(\nabla f \cdot \boldsymbol{ F })
    \nabla \times (f \boldsymbol{ F }) = (\nabla f) \times \boldsymbol{ F } + f(\nabla f \times \boldsymbol{ F })
    \nabla ^2 f = f''(r) + \frac{ 2 }{ r } f'(r)
    \nabla (\boldsymbol{ F } \cdot \boldsymbol{ G }) = (\boldsymbol{ G } \cdot \nabla) \boldsymbol{ F } + (\boldsymbol{ F } \cdot \nabla) \boldsymbol{ G } + \boldsymbol{ F } \times (\nabla \times \boldsymbol{ G }) + \boldsymbol{ G } \times (\nabla \times \boldsymbol{ F })
    \nabla \cdot (\boldsymbol{ F } \times \boldsymbol{ G }) = (\nabla \times \boldsymbol{ F }) - \boldsymbol{ F } \cdot (\nabla \times \boldsymbol{ G })
    \nabla \times (\boldsymbol{ F } \times \boldsymbol{ G }) = (\nabla \cdot \boldsymbol{ G }) \boldsymbol{ F } - (\nabla \cdot \boldsymbol{ F }) \boldsymbol{ G } + (\boldsymbol{ G } \cdot \nabla) \boldsymbol( F ) - (\boldsymbol{ F } \cdot \nabla) \boldsymbol( G )
    \nabla r^α = αr^{α-2} \boldsymbol{ r }
    \Delta r^α = α(α + 1)r^{α - 2}
    \nabla \cdot (r^α \boldsymbol{ r }) = (α + 3)r^α
    \nabla \log r = \frac{ \boldsymbol{ r } }{ r^2 }
    \nabla (\frac{ \boldsymbol{ r } }{ r }) = \frac{ 2 }{ r }
    \Delta (\frac{ 1 }{ r^2 }) = \frac{ 4 }{ r^4 }
    \nabla \times \boldsymbol{ r } = C
    \nabla \cdot (\nabla \times \boldsymbol{ F }) = 0
    \nabla \times (\nabla f) = \boldsymbol{ 0 }
    df = (\nabla f) \cdot d\boldsymbol{ r } = (d\boldsymbol{ r } \cdot \nabla)f
        \nabla \times (\nabla \times \boldsymbol{ F }) = \nabla (\nabla \cdot \boldsymbol{ F }) - \triangle \boldsymbol{ F }

    \Delta \boldsymbol{ F } = \Delta F_x \boldsymbol{ i } + \Delta F_y \boldsymbol{ j } + \Delta F_y \boldsymbol{ k }
    \begin{eqnarray}\Delta f = \nabla^2 f = \frac{ \partial^2 f }{ \partial x^2 }   + \frac{ \partial^2 f }{ \partial y^2 }   + \frac{ \partial^2 f }{ \partial z^2 }\end{eqnarray}

円柱座標

    u = (r, \theta, z)
    \nabla = ( \frac{ \partial }{ \partial r },  \frac{ 1 }{ r } \frac{ \partial }{ \partial \theta }, \frac{ \partial }{ \partial z })
    \nabla \cdot \boldsymbol{ A } = \frac{ 1 }{ r }\frac{ \partial }{ \partial r }(rA_r) + \frac{ 1 }{ r }\frac{ \partial }{ \partial \theta }(A_\theta) + \frac{ \partial }{ \partial z }(A_z)
    \nabla \times \boldsymbol{ A } =
              { \frac{ 1 }{ r } \frac{ \partial }{ \partial \theta }( A_z ) - \frac{ \partial }{ \partial \theta }(A_\theta)} \boldsymbol{ r }
             + { \frac{ \partial }{ \partial z }( A_r ) - \frac{ \partial }{ \partial r }(A_z)} \boldsymbol{ \theta }
             +  \frac{ 1 }{ r } { \frac{ \partial }{ \partial z }( rA_r ) - \frac{ \partial }{ \partial \theta }(A_r)} \boldsymbol{ z }
    \Delta u = u_rr + \frac{ 1 }{ r } u_r + \frac{ 1 }{ r^2 }u_\theta\theta + u_zz
     \boldsymbol{ v } = \dot{ r }\boldsymbol{ r } + r\dot{ \theta }\boldsymbol{ \theta } + \dot{ z }\boldsymbol{ z }
     \boldsymbol{ a } = (\ddot{ r } - r\ddot{ \theta })\boldsymbol{ r } + \frac{ 1 }{ r }\frac{ d }{ dt }(r^2 \dot{ \theta })\boldsymbol{ \theta } + \ddot{ z }\boldsymbol{ z }

球座標

    u = (r, \theta, \phi)
    \Delta = (\frac{ \partial }{ \partial r}, \frac{ 1 }{ r }\frac{ \partial }{ \partial \theta}, \frac{ 1 }{ r\sin \theta })
    \nabla \cdot \boldsymbol{ A } = \frac{ 1 }{ r^2 } \frac{ \partial }{ \partial r }(r^2A_r) + \frac{ 1 }{ r\sin \theta }\frac{ \partial }{ \partial \theta }(A_\theta \sin \theta) + \frac{ 1 }{ r\sin \theta }\frac{ \partial }{ \partial \phi }(A_\phi)
    \nabla \times \boldsymbol{ A } = \frac{ 1 }{ r\sin \theta } { \frac{ \partial }{ \partial \theta}(A_\phi \sin \theta)- \frac{ \partial }{ \partial \theta}}\boldsymbol{ r }
    \frac{ 1 }{ r } {\frac{ 1 }{ \sin \theta }\frac{ \partial }{ \partial \phi }(A_r)- \frac{ \partial }{ \partial r }(rA_\phi)}\boldsymbol{ \theta }
    \frac{ 1 }{ r }{\frac{ \partial }{ \partial r}(rA_\theta)- \frac{ \partial }{ \partial \theta}(A_r)}\boldsymbol{ \theta }
    \boldsymbol{ v } = \dot{ r }\boldsymbol{ r }+ r\dot{ \theta }\boldsymbol{ \theta }+ r\sin \theta \dot{ \phi }\boldsymbol{ \phi }
    \boldsymbol{ a } = {\ddot{ r }- r\dot{\theta}^2- r\sin^2 \theta \dot{\phi}^2}\boldsymbol{ r }
    +{\frac{ 1 }{ r }\frac{ d }{ dt }(r^2 \theta )- r\sin \theta \cos \theta \dot{\phi}^2}\boldsymbol{ \theta }
    +{\frac{ 1 }{ r\sin \theta }\frac{ d }{ dt }(r^2 \sin^2\theta \dot{\phi})}\boldsymbol{ \phi }
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