Mathematical Physics Vol 1

82 Chapter 4. Field theory where α , β and γ are angles between the vector −→ MN and positive directions of axes x , y and z , respectively. Thus, the derivative of a scalar function f in the direction of e , can also be represented as follows

∂ f ∂ x ·

∂ f ∂ y ·

∂ f ∂ z ·

∆ x +

∆ y +

∆ z + δ x · ∆ x + δ y · ∆ y + δ z · ∆ z ∆ s

∆ f ∆ s

d f d s

= lim ∆ s → 0

= lim ∆ s → 0

∂ f ∂ y ·

∂ f ∂ z ·

∂ f ∂ x ·

cos α +

cos β +

cos γ +

=

+ cos α lim ∆ s → 0

δ x + cos β lim ∆ s → 0

δ y + cos γ lim ∆ s → 0

δ z =

∂ f ∂ x ·

∂ f ∂ y ·

∂ f ∂ z ·

cos α +

cos β +

cos γ .

(4.11)

=

The directional derivative can be obtained in the following manner as well. Observe the function f ( x , y , z ) defined in the neighborhood of point M ( a , b , c ) , which lies on direction ℓ . Let this direction be determined by the unit vector e , and let r M and r be the position vectors of points M and N ∈ ℓ , respectively, (Fig. 4.3) ∆ s = MN , r = r M + ∆ s · e , then x = a + ∆ s · cos α , y = b + ∆ s · cos β , z = c + ∆ s · cos γ . (4.12) We further obtain the following for the directional derivative

d f d s

f ( N ) − f ( M ) ∆ s f ( r ) − f ( r M ) ∆ s

D e f =

= lim N → M = lim ∆ s → 0 = lim ∆ s → 0

=

=

f ( r M + ∆ s e ) − f ( r M ) ∆ s .

Figure 4.3: Increment ∆ s .

Converting the function f ( N ) into a power series in the neighbourhood of point M ( a , b , c ) and using (4.12), we obtain f ( N )= f ( x , y , z )= f ( a + ∆ s · cos α , b + ∆ s · cos β , c + ∆ s · cos γ )= (4.13) = f ( M )+ 1 1! ∂ f ∂ x   M · cos α + ∂ f ∂ y   M · cos β + ∂ f ∂ z   M · cos γ · ∆ s + δ ( N ) · ∆ s , where lim N → M δ ( N )= 0, and it thus follows that f ( N ) − f ( M ) ∆ s = (4.14) = ∂ f ∂ x   M · cos α + ∂ f ∂ y   M · cos β + ∂ f ∂ z   M · cos γ + δ ( N ) , that is D e f = d f d s = lim ∆ s → 0 f ( N ) − f ( M ) ∆ s = ∂ f ∂ x   M · cos α + ∂ f ∂ y   M · cos β + ∂ f ∂ z   M · cos γ = = f x · cos α + f y · cos β + f z · cos γ . (4.15)