Mathematical Physics Vol 1

Chapter 4. Field theory

100

Let us further assume that ϕ is an integrable function on the surface S , that is, that there exist an integral over the closed surface S : I = x S ϕ ( r ) ◦ d S . (4.73) This integral can be a scalar or vector function of the value V , of the region V , bordered by the closed surface S . Observe now the value I / V and let the surface "tighten" around the fixed point A , that is, let V → 0. The question now arises as to the existence and determination of the limit value of the quotient I / V .

Definition We call the spatial derivative of the function ϕ ( r ) the limit value

s S ϕ ( r ) ◦ d S V

lim V → 0

(4.74)

if it exists.

If ϕ ( r ) is a scalar position function, then ϕ ( r ) ◦ d S is a vector, and consequently the spatial derivative is also a vector, which will be denoted by

s S ϕ ( r ) d S V .

∇ ϕ ( r )= lim V → 0

(4.75)

It can be proved that this quantity represents the already defined gradient

s S ϕ ( r ) d S V .

grad ϕ = ∇ ϕ ( r )= lim V → 0

(4.76)

If ϕ ( r ) is a vector position function

ϕ ( r ) ≡ v ( r ) ,

(4.77)

then, depending on the meaning of the circle-product, we can distinguish two possible cases. In the first case, where the circle-product represents scalar multiplication, the product v ◦ d S represents a scalar, and consequently the spatial derivative is also a scalar, denoted by

s S v · d S V

∇ v = div v = lim V → 0

(4.78)

which will be called divergence . In the second case, where the circle-product represents vector multiplication, the product v × d S represents a vector, and consequently the spatial derivative is also a vector, denoted by ∇ × v = rot v = lim V → 0 s S v × d S V (4.79) and defining a value called rotor . From previous definitions of gradient , divergence and rotor , their independence from the choice of coordinate system follows, which has already been mentioned when they were defined in the previous chapter.