Mathematical Physics Vol 1

Chapter 4. Field theory

88

a) ∇ ( f · g )= g ∇ f + f ∇ g , b) ∇ · ( f a )=( ∇ f ) · a + f ( ∇ · a ) ,

c) ∇ × ( f a )=( ∇ f ) × a + f ( ∇ × a ) , d) ∇ · ( a × b )=( ∇ × a ) · b − a · ( ∇ × b ) , e) ∇ ( a · b )= a × ( ∇ × b )+ b × ( ∇ × a )+( a · ∇ ) b +( b · ∇ ) a . 6 f) ∇ × ( a × b )=( b · ∇ ) a − b ( ∇ · a ) − ( a · ∇ ) b + a ( ∇ · b ) .

4.1.5 Laplace or delta operator

Let us now define an operator, of scalar nature, as follows ∇ · ∇ = ∇ 2 = = ∂ ∂ x i + ∂ ∂ y j + ∂ ∂ z k · ∂ ∂ x i + ∂ ∂ y j + ∂ ∂ z k =

(4.32)

∂ 2 ∂ x 2

∂ 2 ∂ y 2

∂ 2 ∂ z 2

= ∆ .

=

+

+

The symbol ∆ - "delta", stands for the Laplace operator 7 or Laplacian.

∂ ∂ x

∂ ∂ y

∂ ∂ z

6 Note that a · ∇ = a x

+ a y

+ a z

7 Pierre Simon Marquis De Laplace (1749-1827), great French mathematician. He laid the foundations of potential theory and made major contributions in mechanics, astronomy, as well as in the field of special functions and probability theory. It is interesting to note that Napoleon Bonaparte was also his student for one year.