Vector identities -vector calculus formulas

Vector identities

Addition and multiplication:

1. A+B=B+A 1. \ A + B = B + A
2. A·B=B·A 2. \ A \cdot B = B \cdot A
3. A×B=B×A 3. \ A \times B = -B \times A
4. (A+B)·C=A·C+B·C 4. \ (A + B)\cdot C=A \cdot C +B \cdot C
5. (A+B)×C=A×C+B×C 5. \ (A + B) \times C= A \times C + B \times C
6. A·(B×C)=B·(C×A)=C·(A×B) 6. \ A\cdot(B\times C)=B\cdot(C\times A)=C\cdot(A\times B)
7. A×(B×C)=(A·C)B(A·B)C 7. \ A\times(B\times C)=(A\cdot C)B-(A\cdot B)C
8. (A×B)×C=(A·C)B(B·C)A 8. \ (A\times B)\times C=(A\cdot C)B-(B\cdot C)A
9. A×(B×C)=(A×B)×C+B×(A×C) 9. \ A\times(B\times C)= (A \times B) \times C + B \times (A \times C)
10. A×(B×C)+B×(C×A)+C×(A×B)=0 10. \ A\times(B\times C)+B\times(C\times A)+C\times(A\times B)=0
11. (A×B)·(C×D)=(A·C)(B·D)(A·D)(B·C) 11. \ (A \times B) \cdot (C \times D) = (A\cdot C)(B\cdot D) − (A\cdot D)(B\cdot C)
12. (A×B)×(C×D)=(A·(B×D))C(A·(B×C))D 12. \ (A \times B) \times (C \times D) = (A\cdot (B \times D))C − (A\cdot (B \times C))D

Gradient Function:

1. (f+g)=f+g 1. \ \overrightarrow {\nabla} (f+g)=\overrightarrow {\nabla}f+\overrightarrow {\nabla}g
2. (cf)=c,for a constant c 2. \ \overrightarrow {\nabla} (cf)= c \overrightarrow {\nabla}, \text{for a constant c}
3. (fg)=fg+gf 3. \ \overrightarrow {\nabla} (fg)= f\overrightarrow {\nabla}g+ g\overrightarrow {\nabla}f
4. fg=(gffg)g2 4. \ \overrightarrow {\nabla} \left(\frac fg \right)= \frac {(g\overrightarrow {\nabla}f-f \overrightarrow {\nabla} g)}{g^2}
5. (F·G)=F×(×G)(×F)×G+(G·)F+(F·) 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. ·(F+G)=·F+·G 1. \ \overrightarrow {\nabla} \cdot (\overrightarrow F + \overrightarrow G)=\overrightarrow {\nabla} \cdot \overrightarrow F +\overrightarrow {\nabla} \cdot \overrightarrow G
2. ·(cF)=c·F,for a constant c 2. \ \overrightarrow {\nabla} \cdot (c \overrightarrow F)= c \overrightarrow {\nabla} \cdot \overrightarrow F , \text {for a constant c}
3. ·(fF)=f·F+F· 3. \ \overrightarrow {\nabla} \cdot (f \overrightarrow F)=f \overrightarrow {\nabla} \cdot \overrightarrow F + \overrightarrow F \cdot \overrightarrow {\nabla}
4. ·(F×G)=G·(×F)F·(×G) 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. ×(F+G)=×F+×G 1. \ \overrightarrow {\nabla} \times (\overrightarrow F + \overrightarrow G)=\overrightarrow {\nabla} \times \overrightarrow F + \overrightarrow {\nabla} \times \overrightarrow G
2. ×(cF)=c×F,for a constant c 2. \ \overrightarrow {\nabla} \times (c \overrightarrow F)= c \overrightarrow {\nabla} \times \overrightarrow F , \text {for a constant c}
3. ×(fF)=f×F+f×F 3. \ \overrightarrow {\nabla} \times (f \overrightarrow F)=f \overrightarrow {\nabla} \times \overrightarrow F + \overrightarrow {\nabla} f \times \overrightarrow F
4. ×(F×G)=F·(·G)(·F)G+(G·)F(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. 2(f+g)=2f+2g 1. \ \overrightarrow {\nabla^2} (f+g)=\overrightarrow {\nabla^2} f+\overrightarrow {\nabla^2} g
2. 2(cf)=c2f,for a constant c 2. \ \overrightarrow {\nabla^2} (cf)= c \overrightarrow {\nabla^2}f , \text {for a constant c}
3. 2(fg)=f2g+2f·g+g2 3. \ \overrightarrow {\nabla^2} (fg)=f\overrightarrow {\nabla^2}g+2\overrightarrow {\nabla}f \cdot g+g\overrightarrow {\nabla^2}

Degree Two Function:

1. ·(×F)=0 1. \ \overrightarrow {\nabla} \cdot (\overrightarrow {\nabla} \times \overrightarrow F)=0
2. ×(f)=0 2. \ \overrightarrow {\nabla} \times (\overrightarrow {\nabla} f)=0
3. ·(f×g)=0 3. \ \overrightarrow {\nabla} \cdot (\overrightarrow {\nabla} f \times \overrightarrow {\nabla} g)=0
4. ·(fggf)=f2gg2f 4. \ \overrightarrow {\nabla} \cdot (f\overrightarrow {\nabla} g − g\overrightarrow {\nabla} f)=f\overrightarrow {\nabla^2}g−g\overrightarrow {\nabla^2}f
5. ×(×F)=(·F)2 5. \ \overrightarrow {\nabla} \times (\overrightarrow {\nabla} \times \overrightarrow F)=\overrightarrow {\nabla}(\overrightarrow {\nabla} \cdot \overrightarrow F)−\overrightarrow {\nabla^2}
6. ·(ψ)=2ψ 6. \ \nabla \cdot (\nabla \psi)= \nabla^2 \psi
7. (·A)×(×A)=2A 7. \ \nabla ( \nabla \cdot A)- \nabla \times (\nabla \times A) = \nabla^2 A
8. ·(ϕψ)=ϕ2ψ+ϕ·ψ 8. \ \nabla \cdot (\phi \nabla \psi ) = \phi \nabla^2 \psi + \nabla \phi \cdot \nabla \psi
9. ψ2ϕϕ2ψ=·(ψϕϕψ) 9. \ \psi \nabla^2 \phi-\phi \nabla^2 \psi= \nabla \cdot (\psi \nabla \phi-\phi \nabla \psi)
10. 2(ϕψ)=ϕ2ψ+2(ϕ)·(ψ)+(2ϕ)ψ 10. \ \nabla^2 (\phi\psi)=\phi \nabla^2 \psi + 2 (\nabla \phi) \cdot (\nabla \psi) + (\nabla^2 \phi)\psi
11. 2(ψA)=A2ψ+2(ψ·)A+ψ2A 11. \ \nabla^2 (\psi A)=A \nabla^2 \psi +2 (\nabla \psi \cdot \nabla)A+ \psi \nabla^2 A
12. 2(A·B)=A·2BB·2A+2·((B·)A+B×(×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. 2(ψ)=(·(ψ))=(2ψ) 1. \ \nabla^2(\nabla \psi)= \nabla (\nabla \cdot (\nabla \psi))=\nabla(\nabla^2 \psi)
2. 2(·A)=·((·A))=·(2A) 2. \ \nabla^2(\nabla \cdot A)= \nabla \cdot (\nabla(\nabla \cdot A))=\nabla \cdot (\nabla^2 A)
3. 2(×A)=×(×(×A))=×(2A) 3. \ \nabla^2(\nabla \times A)= -\nabla \times (\nabla \times (\nabla \times A))=\nabla \times (\nabla^2 A)

 

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