Mathematical Physics Vol 1

4.1 Scalar field

81

Definition If the limit value

∆ f ∆ s

f ( N ) − f ( M ) ∆ s

lim ∆ s → 0 (4.5) exists, then it is called the derivative of the function f , at point M, in the direction of e , and denoted by D e f = d f d s . = lim ∆ s → 0 , It is obvious that this derivative represents the rate of change of function f , at point M , i the direction of e . Both ways of denoting, D e f or d f / d s , are common, but D e f is more convenient as it indicates the direction of change. The directional derivative of a function, as per definition (4.5), does not depend on the choice of the coordinate system. However, in order to calculate specific values of these derivatives, they will be observed with respect to, for example, Cartesian coordinate system, and f ( M ) will be represented as a function of f ( x , y , z ) . To this end, we will observe the change in the function f while moving from point M ( x , y , z ) in a given field to a close point N ( x + ∆ x , y + ∆ y , z + ∆ z ) in the same field in the direction of e . Further, based on the definition of the partial derivative of a scalar function with respect to, for example, x , we obtain:

∂ f ∂ x

f ( x + ∆ x , y , z ) − f ( x , y , z ) ∆ x

∆ f x ∆ x

= lim

= lim

(4.6)

,

∆ x → 0

∆ x → 0

which can then be rewritten, in case of a differentiable function, as

∂ f ∂ x · ∆ x + δ x · ∆ x , gde δ x → 0 kada ∆ x → 0 .

∆ f x =

(4.7)

The two remaining increments ∆ f y and ∆ f z are obtained in the same way, and thus the total increment of the function f can be expressed as ∆ f = f ( N ) − f ( M )= (4.8)

∂ f ∂ x ·

∂ f ∂ y ·

∂ f ∂ z ·

∆ x +

∆ y +

∆ z + δ x · ∆ x + δ y · ∆ y + δ z · ∆ z .

=

Further, in Cartesian coordinates ∆ s = p ∆ x 2 + ∆ y 2 + ∆ z 2 . (4.9) Assume now that ∆ s → 0, iff ∆ x → 0, ∆ y → 0 i ∆ z → 0. According to Fig. 4.2,

Figure 4.2: Increments in the directions of x and y axes.

it is obvious that

∆ x ∆ s

∆ y ∆ s

∆ z ∆ s

= cos α ,

= cos β , and by analogy

= cos γ ,

(4.10)