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• Cartesiano ($$x$$,$$y$$,$$z$$)
• cilíndrico ($$\rho$$,$$\phi$$,$$z$$)
• esférico ($$r$$,$$\theta$$,$$\phi$$)

Los vectores de base asociados se identifican usando un caret ($$\hat{~}$$) sobre el símbolo. El operando vectorial$${\bf A}$$ se expresa en términos de componentes en las direcciones de base de la siguiente manera:

• Cartesiano:$${\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z$$
• Cilíndrico$${\bf A} = \hat{\bf \rho}A_{\rho} + \hat{\bf \phi}A_{\phi} + \hat{\bf z}A_z$$
• esférico:$${\bf A} = \hat{\bf r}A_r + \hat{\bf \theta}A_{\theta} + \hat{\bf \phi}A_{\phi}$$

\ begin {align}\ nabla f &=\ hat {\ bf x}\ frac {\ parcial f} {\ parcial x} +\ hat {\ bf y}\ frac {\ parcial f} {\ parcial y} +\ hat {\ bf z}\ frac {\ parcial f} {\ parcial z} &\ end {align}

\ begin {align}\ nabla f &=\ hat {\ bf\ rho}\ frac {\ parcial f} {\ parcial\ rho} +\ hat {\ bf\ phi}\ frac {1} {\ rho}\ frac {\ parcial f} {\ parcial\ phi} +\ hat {\ bf z}\ frac {\ parcial f} {\ parcial z} &\ end {alinear}

\ begin {align}\ nabla f &=\ hat {\ bf r}\ frac {\ parcial f} {\ parcial r} +\ hat {\ bf\ theta}\ frac {1} {r}\ frac {\ parcial f} {\ parcial\ theta} +\ sombrero {\ bf\ phi}\ frac {1} {r\ sin\ theta}\ frac {parcial\ f} {\ parcial\ phi} &\ end {align}

## Divergencia

\ begin {align}\ nabla\ cdot {\ bf A} &=\ frac {\ parcial a_x} {\ parcial x} +\ frac {\ parcial a_Y} {\ parcial y} +\ frac {\ parcial a_Z} {\ parcial z} &\ end {align}

\ begin {align}\ nabla\ cdot {\ bf A} &=\ frac {1} {\ rho}\ frac {\ parcial} {\ parcial\ rho}\ izquierda (\ rho A_ {\ rho}\ rho}\ derecha) +\ frac {1} {\ rho}\ frac {\ parcial A_ {\ phi}} {\ parcial\ phi} +\ frac {\ parcial a_Z} {\ z parcial} &\ end {align}

\ begin {align}\ nabla\ cdot {\ bf A} &= ~~\ frac {1} {r^2}\ frac {\ parcial} {\ parcial} {\ parcial}\ izquierda (r^2 a_R\ derecha) &\ nonumber\\ &~~ +\ frac {1} {r\ sin\ theta}\ frac {\ parcial} {\ parcial\ theta} izquierda (A_ {\ theta}\ sin\ theta\ derecha) &\ nonumber\\ &~~ +\ frac {1} {r\ sin\ theta}\ frac {\ parcial A_ {\ phi}} {\ parcial\ phi} & \ end {align}

## Curl

\ begin {align}\ nabla\ times {\ bf A} &= ~~\ hat {\ bf x}\ izquierda (\ frac {\ parcial a_z} {\ parcial y} -\ frac {\ parcial a_y} {\ parcial z}\ derecha) &\ nonumber\\ &~~ +\ hat {\ bf y}\ izquierda (\ frac {\ parcial a_x} {parcial\ z} -\ frac {\ parcial a_Z} {\ parcial x}\ derecha) &\ nonumber\\ &~~ +\ hat {\ bf z}\ izquierda (\ frac {\ A_y parcial} {\ x parcial} -\ frac {\ parcial a_x} {\ parcial y}\ derecha) &\ etiqueta {m0139_ecurlCart}\ end {align}

\ begin {align}\ nabla\ times {\ bf A} &= ~~\ hat {\ bf\ rho}\ left (\ frac {1} {\ rho}\ frac {\ parcial a_Z} {\ parcial\ phi} -\ frac {\ parcial A_ {\ phi}} {\ parcial z}\ derecha) &\ nonumber\\ &~~ +\ hat {\ bf\ phi}\ izquierda (\ frac {\ parcial A_ {\ rho}} {\ z parcial} -\ frac {\ parcial a_Z} {\ parcial\ rho}\ derecha) &\ nonumber\\ & amp; ~~ +\ hat {\ bf z}\ frac {1} {\ rho}\ izquierda [\ frac {\ parcial} {\ parcial\ rho}\ izquierda (\ rho A_ {\ phi}\ derecha) -\ frac {\ parcial A_ {\ rho}} {\ parcial\ phi}\ derecha] &\ end {align}

\ begin {align}\ nabla\ times {\ bf A} &= ~~\ hat {\ bf r}\ frac {1} {r\ sin\ theta}\ izquierda [\ frac {\ parcial} {\ parcial\ theta}\ izquierda (A_ {\ phi}\ sin\ theta\ derecha) -\ frac {\ parcial A_ {\ theta}} {\ phi parcial\}\ derecha] &\ nonumber\\ &~~ +\ sombrero {\ bf\ theta}\ frac {1} {r}\ izquierda [\ frac {1} {\ sin\ theta}\ frac {\ parcial a_R} {\ parcial\ phi} -\ frac {\ parcial} {\ r parcial}\ izquierda (rA_ {\ phi}\ derecha)\ derecha] &\ nonumber\\ &~~ +\ hat {\ bf\ phi}\ frac {1} {r}\ izquierda [\ frac {\ parcial} {\ parcial}\ izquierda (r A_ {\ theta}\ derecha) -\ frac {parcial\ a_R} {\ parcial\ theta}\ derecha]\ etiqueta {m0139_ecurlSPh} &\ end {align}