Mathematical Physics Vol 1

Chapter 4. Field theory

92

Definition A vector function, defined by

i

j k

∂ ∂ x ∂ ∂ z v x v y v z ∂ ∂ y

rot v = ∇ × v =

(4.46)

=

=

∂ z

i +

∂ v z ∂ x

j +

∂ v x ∂ y

∂ v y

∂ v y ∂ x −

∂ v z ∂ y −

∂ v x ∂ z −

k

is called the rotor of the vector function v or the rotor of the vector field, defined by the function v .

Theorem8 The magnitude and direction of the vector rot v do not depend on the specifically selected Cartesian coordinate system.

This statement will be proved later.

4.2.3 Classification of vector fields

Definition A vector field, in which for all its points the following is true rot v = 0 , div v̸ = 0 ,

(4.47)

is called a potential or irrotational or lamellar field.

Definition A vector field, in which for all its points the following is true rot v̸ = 0 , div v = 0 ,

(4.48)

is called a solenoidal or rotational field.

Definition A vector field, in which for all its points the following is true rot v = 0 , div v = 0 ,

(4.49)

is called a Laplace field .