Vector identities

Addition and multiplication:

$$ 1. \ A + B = B + A $$

$$ 2. \ A \cdot B = B \cdot A $$

$$ 3. \ A \times B = -B \times A $$

$$ 4. \ (A + B)\cdot C=A \cdot C +B \cdot C $$

$$ 5. \ (A + B) \times C= A \times C + B \times C $$

$$ 6. \ A\cdot(B\times C)=B\cdot(C\times A)=C\cdot(A\times B) $$

$$ 7. \ A\times(B\times C)=(A\cdot C)B-(A\cdot B)C $$

$$ 8. \ (A\times B)\times C=(A\cdot C)B-(B\cdot C)A $$

$$ 9. \ A\times(B\times C)= (A \times B) \times C + B \times (A \times C) $$

$$ 10. \ A\times(B\times C)+B\times(C\times A)+C\times(A\times B)=0 $$

$$ 11. \ (A \times B) \cdot (C \times D) = (A\cdot C)(B\cdot D) − (A\cdot D)(B\cdot C) $$

$$ 12. \ (A \times B) \times (C \times D) = (A\cdot (B \times D))C − (A\cdot (B \times C))D $$

Gradient Function:

$$ 1. \ \overrightarrow {\nabla} (f+g)=\overrightarrow {\nabla}f+\overrightarrow {\nabla}g $$

$$ 2. \ \overrightarrow {\nabla} (cf)= c \overrightarrow {\nabla}, \text{for a constant c} $$

$$ 3. \ \overrightarrow {\nabla} (fg)= f\overrightarrow {\nabla}g+ g\overrightarrow {\nabla}f $$

$$ 4. \ \overrightarrow {\nabla} \left(\frac fg \right)= \frac {(g\overrightarrow {\nabla}f-f \overrightarrow {\nabla} g)}{g^2} $$

$$ 5. \ \overrightarrow {\nabla} (\overrightarrow F \cdot \overrightarrow G) = \overrightarrow F \times (\overrightarrow {\nabla} \times \overrightarrow G) - (\overrightarrow {\nabla} \times \overrightarrow F) \times \overrightarrow G + (\overrightarrow G \cdot \overrightarrow {\nabla} )\overrightarrow F + (\overrightarrow F \cdot \overrightarrow {\nabla}) $$

Divergence Function:

$$ 1. \ \overrightarrow {\nabla} \cdot (\overrightarrow F + \overrightarrow G)=\overrightarrow {\nabla} \cdot \overrightarrow F +\overrightarrow {\nabla} \cdot \overrightarrow G $$

$$ 2. \ \overrightarrow {\nabla} \cdot (c \overrightarrow F)= c \overrightarrow {\nabla} \cdot \overrightarrow F , \text {for a constant c} $$

$$ 3. \ \overrightarrow {\nabla} \cdot (f \overrightarrow F)=f \overrightarrow {\nabla} \cdot \overrightarrow F + \overrightarrow F \cdot \overrightarrow {\nabla} $$

$$ 4. \ \overrightarrow {\nabla} \cdot (\overrightarrow F \times \overrightarrow G)=G \cdot (\overrightarrow {\nabla} \times \overrightarrow F)−\overrightarrow F \cdot (\overrightarrow {\nabla} \times \overrightarrow G) $$

Curl Function:

$$ 1. \ \overrightarrow {\nabla} \times (\overrightarrow F + \overrightarrow G)=\overrightarrow {\nabla} \times \overrightarrow F + \overrightarrow {\nabla} \times \overrightarrow G $$

$$ 2. \ \overrightarrow {\nabla} \times (c \overrightarrow F)= c \overrightarrow {\nabla} \times \overrightarrow F , \text {for a constant c} $$

$$ 3. \ \overrightarrow {\nabla} \times (f \overrightarrow F)=f \overrightarrow {\nabla} \times \overrightarrow F + \overrightarrow {\nabla} f \times \overrightarrow F$$

$$ 4. \ \overrightarrow {\nabla} \times (\overrightarrow F \times \overrightarrow G)=\overrightarrow F \cdot (\overrightarrow {\nabla} \cdot \overrightarrow G)−(\overrightarrow {\nabla} \cdot \overrightarrow F)\overrightarrow G +(\overrightarrow G \cdot \overrightarrow {\nabla})\overrightarrow F − (\overrightarrow F \cdot \overrightarrow {\nabla}) $$

Laplacian Function:

$$ 1. \ \overrightarrow {\nabla^2} (f+g)=\overrightarrow {\nabla^2} f+\overrightarrow {\nabla^2} g $$

$$ 2. \ \overrightarrow {\nabla^2} (cf)= c \overrightarrow {\nabla^2}f , \text {for a constant c}$$

$$ 3. \ \overrightarrow {\nabla^2} (fg)=f\overrightarrow {\nabla^2}g+2\overrightarrow {\nabla}f \cdot g+g\overrightarrow {\nabla^2} $$

Degree Two Function:

$$ 1. \ \overrightarrow {\nabla} \cdot (\overrightarrow {\nabla} \times \overrightarrow F)=0 $$

$$ 2. \ \overrightarrow {\nabla} \times (\overrightarrow {\nabla} f)=0 $$

$$ 3. \ \overrightarrow {\nabla} \cdot (\overrightarrow {\nabla} f \times \overrightarrow {\nabla} g)=0 $$

$$ 4. \ \overrightarrow {\nabla} \cdot (f\overrightarrow {\nabla} g − g\overrightarrow {\nabla} f)=f\overrightarrow {\nabla^2}g−g\overrightarrow {\nabla^2}f $$

$$ 5. \ \overrightarrow {\nabla} \times (\overrightarrow {\nabla} \times \overrightarrow F)=\overrightarrow {\nabla}(\overrightarrow {\nabla} \cdot \overrightarrow F)−\overrightarrow {\nabla^2} $$

$$ 6. \ \nabla \cdot (\nabla \psi)= \nabla^2 \psi $$

$$ 7. \ \nabla ( \nabla \cdot A)- \nabla \times (\nabla \times A) = \nabla^2 A $$

$$ 8. \ \nabla \cdot (\phi \nabla \psi ) = \phi \nabla^2 \psi + \nabla \phi \cdot \nabla \psi $$

$$ 9. \ \psi \nabla^2 \phi-\phi \nabla^2 \psi= \nabla \cdot (\psi \nabla \phi-\phi \nabla \psi) $$

$$ 10. \ \nabla^2 (\phi\psi)=\phi \nabla^2 \psi + 2 (\nabla \phi) \cdot (\nabla \psi) + (\nabla^2 \phi)\psi $$

$$ 11. \ \nabla^2 (\psi A)=A \nabla^2 \psi +2 (\nabla \psi \cdot \nabla)A+ \psi \nabla^2 A $$

$$ 12. \ \nabla^2 (A \cdot B)=A \cdot \nabla^2B - B \cdot \nabla^2A+ 2 \nabla \cdot ((B \cdot \nabla)A+B \times (\nabla \times A)) $$

Third derivatives:

$$ 1. \ \nabla^2(\nabla \psi)= \nabla (\nabla \cdot (\nabla \psi))=\nabla(\nabla^2 \psi)$$

$$ 2. \ \nabla^2(\nabla \cdot A)= \nabla \cdot (\nabla(\nabla \cdot A))=\nabla \cdot (\nabla^2 A) $$

$$ 3. \ \nabla^2(\nabla \times A)= -\nabla \times (\nabla \times (\nabla \times A))=\nabla \times (\nabla^2 A) $$