Mathematical Physics Vol 1

4.2 Vector field

101

Divergence and rotor of the vector function of a constant direction It is of special interest to find the expression for the divergence of the vector function of a constant direction. Observe one such vector c c = c c 0 , where c 0 is the unit vector of a constant direction . (4.80) Further, according to the definition (4.81) Using this relation, we can write an analytical expression for the divergence in a rectangular coordinate system. Observe a vector function, expressed in one of the possible ways 10 v = v x i + v y j + v z k = v 1 e 1 + v 2 e 2 + v 3 e 3 = v i e i . (4.82) According to the relation above div c = lim V → 0 s S c d S V = lim V → 0 s S c d S V · c 0 = grad c · c 0 .

div v = div ( v x i )+ div ( v y j )+ div ( v z k )= = i · grad v x + j · grad v y + k · grad v z =

(4.83)

3 ∑ i = 1

e i · grad v i ≡ e i · grad v i .

=

As

∂ v i ∂ x ∂ v x ∂ x

∂ v i ∂ y ∂ v y ∂ y

∂ v i ∂ z

i +

j +

k ,

grad v i =

(4.84)

we finally obtain

∂ v z ∂ z

div v = (4.85) Thus, the same expression as in the previous chapter for Cartesian coordinates. In the case of rotor, it is useful to determine it for the vector of a constant direction c = c c 0 , c = | c | c 0 = −−−→ const. (4.86) From the definition we obtain rot c = lim V → 0 s S c × d S V = lim V → 0 s S c d S V × c 0 = grad c × c 0 . (4.87) If we apply this to some vector function a = a x i + a y j + a z k , (4.88) we obtain + + .

i

j k

∂ ∂ x ∂ ∂ z a x a y a z ∂ ∂ y

rot a =

=

(4.89)

=( grad a x × i )+( grad a y × j )+( grad a z × k )= = ∂ a z ∂ y − ∂ a y ∂ z i + ∂ a x ∂ z − ∂ a z ∂ x j + ∂ a y ∂ x − ∂ a x ∂ y k ,

thus, the same expression as in the previous chapter for Cartesian coordinates. 10 Note that when writing this expression, we used the addition convention, according to which ∑ 3

i = 1 v i e i ≡ v i e i , i.e.

addition by repeated indices is performed.