Mathematical Physics Vol 1

4.1 Scalar field

83

Thus, we obtain the same result as in relation (4.11). As an infinite number of directions pass through each point, an infinite number of derivatives can be obtained in those directions. However, if we observe a coordinate system, for example, Cartesian, each of these derivatives can be expressed by a first partial derivative of the function f at point M as follows. Let point M be determined by the position vector r M and let e be a unit vector. Let now ℓ be the line through point M , which can be represented as follows r ( s )= x ( s ) i + y ( s ) j + z ( s ) k = r M + s e ( s ≥ 0 , | e | = 1 ) , (4.16) where r ( s ) is the position vector, depending on parameter s (arc length). Observe the derivative of the function f along the curved line ℓ , then D e f = d f d s is the derivative of the function f [ x ( s ) , y ( s ) , z ( s )] , which depends on the length of the arc s .

Thus, assuming that f has continuous partial deriva tives, and applying the rule on derivation of complex functions, we obtain

∂ f ∂ x · ∂ f ∂ x ·

∂ f ∂ y ·

∂ f ∂ z ·

d f d s

d x d s

d y d s

d z d s

D e f =

=

+

+

=

∂ f ∂ x ·

∂ f ∂ x ·

x ′ +

y ′ +

z ′ ,

(4.17)

=

Figure 4.4: Line ℓ on surface S and the tangent plane at point M .

where ( ′ ) denotes the derivative with respect to the parameter s .

Differentiating the vector function r ( s ) , we obtain from (4.16) 2 : d r d s = t = x ′ i + y ′ j + z ′ k = e .

(4.18)

Thus, in this case e has the direction of the tangent. Bearing in mind the scalar product and the relation (4.18), the expression (4.17) can be transformed as follows D e f = ∂ f ∂ x · i + ∂ f ∂ y · j + ∂ f ∂ z · k x ′ i + y ′ j + z ′ k = = ∂ f ∂ x · i + ∂ f ∂ y · j + ∂ f ∂ z · k e , which leads us to the introduction of the following vector. 2 Let the curved line ℓ be given in the parametric form r ( s )= x ( s ) i + y ( s ) j + z ( s ) k , where the arc of the curve s is the parameter. Then

d r d s

d x d s

d y d s

d y d s

i +

j +

k = t ,

=

where t is the directional vector of unit magnitude of the tangent at a point of the curved line ℓ , as

= s

= | t | = s

d x d s

+

d y d s

+

d z d s

2

2

2

d x 2 + d y 2 + d z 2 d s 2

d r d s

= 1 .