List of vector identities

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This article lists a few helpful mathematical identities which are useful in vector algebra.

1 Triple products
2 Other products
3 Product rules
4 Green's first identity
5 Fundamental theorems
6 See also

[edit] Triple products

$\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{C} \times \vec{B}) \times \vec{A} = \vec{B}(\vec{A} \cdot \vec{C}) - \vec{C}(\vec{A} \cdot \vec{B})$
$\vec{A}\cdot(\vec{B}\times \vec{C}) = \vec{B}\cdot(\vec{C}\times \vec{A}) = \vec{C}\cdot(\vec{A}\times \vec{B})$

these can be proved by giving arbitrary components to $A,$ $B,$ and $C,$

$\vec{A} = (a_x, a_y, a_z)$

$\vec{B} = (b_x, b_y, b_z)$

$\vec{C} = (c_x, c_y, c_z)$

then finding the values of each statement, such as $\vec{B}\times \vec{C}$ , in terms of the generic components will show that both sides of the equation are equal.

[edit] Other products

$(\vec{A} \times \vec{B}) \cdot (\vec{A} \times \vec{B}) = A^2 B^2 - (\vec{A} \cdot \vec{B})^2 = \vec{B} \cdot \vec{A} \times (\vec{B} \times \vec{A})$ (note: this is another form of Lagrange's formula).

[edit] Product rules

$\vec{\nabla} (fg) = f(\vec{\nabla}g) + g(\vec{\nabla} f)$
$\vec{\nabla}(\vec{A} \cdot \vec{B}) = \vec{A} \times (\vec{\nabla} \times \vec{B})+\vec{B} \times (\vec{\nabla} \times \vec{A})+(\vec{A} \cdot \vec{\nabla})\vec{B}+(\vec{B} \cdot \vec{\nabla})\vec{A}$
$\vec{\nabla} \cdot (f\vec{A})=f(\vec{\nabla} \cdot \vec{A})+\vec{A} \cdot (\vec{\nabla} f)$
$\vec{\nabla} \cdot (\vec{A} \times \vec{B})=\vec{B} \cdot (\vec{\nabla} \times \vec{A})-\vec{A} \cdot (\vec{\nabla} \times \vec{B})$
$\nabla\times (\vec{A}\times\vec{B})= (\vec{B}\cdot\nabla) \vec{A}-(\vec{A}\cdot\nabla)\vec{B} + \vec{A} (\nabla\cdot\vec{B}) - \vec{B}(\nabla\cdot\vec{A})$
$\nabla\times (\vec{A}\times\vec{B})= \vec{A} \times (\nabla\times\vec{B}) - \vec{B} \times (\nabla\times\vec{A}) - (\vec{A}\times\nabla) \times \vec{B} + (\vec{B}\times\nabla) \times \vec{A}$
$\nabla\times (f\vec{A})=f(\nabla\times\vec{A})+(\nabla f)\times\vec{A}$

[edit] Green's first identity

$\vec{\nabla} \cdot \left( f \vec{\nabla} f \right) = f \vec{\nabla} \cdot \left( \vec{\nabla} f \right) + \left( \vec{\nabla} f \cdot \vec{\nabla} f \right) = f \nabla^2 f + \left( \vec{\nabla} f \cdot \vec{\nabla} f \right)$
therefore
$f \nabla^2 f = \vec{\nabla} \cdot \left( f \vec{\nabla} f \right) - \left( \vec{\nabla} f \cdot \vec{\nabla} f \right)$