Mathematical Physics Vol 1

Chapter 4. Field theory

86

Thus, when a scalar function is differentiable, then its total differential is equal to the scalar product of the gradient of the function and the position vector differential. This indicates one way to calculate the gradient of a function. Namely, when a function differential can be represented as a scalar product of two vectors, one of which is d r , then the other factor is equal to the gradient of the function.

4.1.2 Partial gradient of a scalar function

Observe a scalar function f depending on two vectors u and v :

f = f ( u, v ) ,

(4.27)

where these two vectors can be represented, with respect to the Cartesian coordinate system, as follows

u = u 1 e 1 + u 2 e 2 + u 3 e 3 , v = v 1 e 1 + v 2 e 2 + v 3 e 3 .

The field of this function is determined by points U and V , which represent the end points of vectors u and v . We used here the following symbols: e 1 = i , e 2 = j and e 3 = k . Let us assume that the vector v is constant and determine the gradient of the function f under this condition. According to (4.19) we obtain

∂ f ∂ u 1

∂ f ∂ u 2

∂ f ∂ u 3

grad u f ( u, v )=

e 1 +

e 2 +

e 3 ,

(4.28)

which is called the partial gradient of the scalar function f , along the vector u . Similarly, we define the partial gradient along the other vector. We can also extend this definition to an arbitrary, but finite, number of vectors. In this case all vectors except one would be considered constant.

4.1.3 Properties of gradient

a) grad C = 0 ( C = const . ) , b) grad ( U + V )=grad U +grad V , gde je U = U ( M ) , V = V ( M ) ,

c) grad ( U · V )= V · grad U + U · grad V , d) grad (C U )=Cgrad U , C=const., e) grad ( U / V )= ( 1 / V 2 )( V · grad U - U · grad V ), f) grad f ( U )= f ′ U · grad U . Based on these properties, the following is true

grad ( C 1 U 1 + C 2 U 2 )= C 1 grad U 1 + C 2 grad U 2 .

(4.29)

Operators having this property are called linear operators . Thus, the gradient is a linear operator.

Proof

a)

grad C =

k C =

∂ ∂ x

∂ ∂ y

∂ ∂ z

∂ C ∂ x

∂ C ∂ y

∂ C ∂ z

i +

j +

i +

j +

k = 0 + 0 + 0 = 0 .