من أجل مجال سلّمي f f f A → \vect A A ∇ → = ( ∂ x , ∂ y , ∂ z ) \vect\nabla = (\partial_x, \partial_y,
\partial_z) ∇ = ( ∂ x , ∂ y , ∂ z )
g r a d → f = ∇ → f = ( ∂ x f , ∂ y f , ∂ z f ) , div A → = ∇ → ⋅ A → = ∂ x A x + ∂ y A y + ∂ z A z , c u r l → A → = ∇ → ∧ A → = ( ∂ y A z − ∂ z A y , ∂ z A x − ∂ x A z , ∂ x A y − ∂ y A x ) , Δ f = div g r a d → f = ∂ x 2 f + ∂ y 2 f + ∂ z 2 f , \begin{align*}
\operatorname{\vect{grad}}f &= \vect\nabla f = (\partial_xf,\ \partial_yf,\ \partial_zf) ,\\
\operatorname{div}\vect A &= \vect\nabla\cdot\vect A = \partial_xA_x + \partial_yA_y + \partial_zA_z ,\\
\operatorname{\vect{curl}}\vect A &= \vect\nabla\wedge\vect A = (\partial_yA_z - \partial_zA_y,\ \partial_zA_x - \partial_xA_z,\ \partial_xA_y - \partial_yA_x) ,\\
\Delta f &= \operatorname{div}\operatorname{\vect{grad}}f = \partial_x^2f + \partial_y^2f + \partial_z^2f ,
\end{align*} grad f div A curl A Δ f = ∇ f = ( ∂ x f , ∂ y f , ∂ z f ) , = ∇ ⋅ A = ∂ x A x + ∂ y A y + ∂ z A z , = ∇ ∧ A = ( ∂ y A z − ∂ z A y , ∂ z A x − ∂ x A z , ∂ x A y − ∂ y A x ) , = div grad f = ∂ x 2 f + ∂ y 2 f + ∂ z 2 f , و Δ A → \Delta\vect A Δ A g r a d → f ⋅ d l → = d f \operatorname{\vect{grad}}f
\cdot\dd\vect l = \dd f grad f ⋅ d l = d f f f f d l → \dd\vect l d l div A → \operatorname{div}
\vect A div A A → \vect A A c u r l → A → ⋅ n → \operatorname{\vect{curl}}\vect A\cdot\vect n curl A ⋅ n A → \vect A A n → \vect n n
Coach