Mathematical Physics Vol 1

Chapter 4. Field theory

84

Definition A vector determined by the relation

∂ f ∂ y

∂ f ∂ z

∂ f ∂ x

i +

j +

k .

grad f =

(4.19)

is called the gradient of the scalar function f .

As the gradient of a scalar function is a vector value, it follows for this vector that magnitude: | grad f | = s ∂ f ∂ x 2 + ∂ f ∂ y 2 + ∂ f ∂ z 2 ,

(4.20)

∂ f / ∂ y | grad f |

∂ f / ∂ z | grad f |

∂ f / ∂ x | grad f |

, cos β =

, cos γ =

direction: cos α =

(4.21)

.

The directional derivative can now be represented as

d f d s

= e · grad f .

D e f =

(4.22)

The geometric interpretation of this product is the projection of the gradient on the direction determined by vector e , that is d f d s = proj e grad f = | grad f |· cos ϕ , (4.23) where ϕ is the angle between grad f and e . It follows from this definition that d f /d s has maximum value when cos ϕ = 1 ⇒ ϕ = 0 . Thus, the scalar field changes most rapidly in the direction of the gradient, i.e. the gradient determines the direction in which the scalar field changes most rapidly.

Figure 4.5: Gradient of function f .

In the special case, when the derivative is obtained in the direction of the + Ox axis, then e=i , and we obtain D i f = i · grad f = i · ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k = ∂ f ∂ x i · i = ∂ f ∂ x . (4.24)