Mathematical Physics Vol 1

4.2 Vector field

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In order to obtain the equation of this surface, let us denote by S the area of the side of the tube, and by d S the vector element of this surface. According to the definition of the solenoid 8 it follows that the vector surface element is orthogonal to the vector v , and thus v · d S = 0 . (4.41) Consequently, if S is the total area of the side, then Z S v · d S = 0 . (4.42) 4.2.2 Divergence and rotor Observe a differentiable vector function v , which can be represented, with respect to the Cartesian coordinate system, in the following form

v = v x i + v y j + v z k .

(4.43)

Definition A scalar function, defined by the relation

∂ v y ∂ y

∂ v x ∂ x

∂ v z ∂ z

div v =

(4.44)

+

+

is called the divergence of the vector function v or divergence of the vector field defined by v .

A more suitable form for denoting divergence is by way of the aforementioned ∇ operator, as follows div v = ∇ · v = = ∂ ∂ x i + ∂ ∂ y j + ∂ ∂ z k · ( v x i + v y j + v z k )= (4.45) = ∂ v x ∂ x + ∂ v y ∂ y + ∂ v z ∂ z .

Theorem7 The value div v depends only of the points in space (and naturally of the value of the function v ), but not on the choice of the coordinate system.

R Note that it is possible to define divergence in such a way that it is obvious that it does not depend on the choice of the coordinate system, which will be done later.

8 Solenoid - form Greek word σωλην - tube.