Lecture ¹14

# Lecture ¹14. The elements of a field‘s theory:

scalar field; vector field; flow(stream), divergence, circulation, curl(swirl); vector interpretation of Ostrogradsky formula.

Exercises for Lecture #14 Examples

The purpose of work:

The acquaintance with the elements of a field‘s theory.

Theoretical information and examples

Content

2.12 Fundamentals of Field Theory

2.12.1 Scalar and vector fields

2.12.2 Gradient

2.12.3 Divergence and rotation

2.12.4 Flux of a vector

2.12 Fundamentals of Field Theory

2.12.1 Scalar and vector fields. A scalar field is defined by the scalar function of the point u = f(P) = f(x, y, z), where P(x, y, z) is a point in space. The surfaces, where C = const, are called level surfaces of the scalar field.

A vector field is defined by the vector function of fhe point a =a(P) = a(r), where P is a point of space and r=xi+yj+zk is the radius vector of the point P. In coordinate form, a = axi + ayj + azk, where ax=ax(x, y, z), ay=ay(x, y, z), az=az(x, y, z) are projections of the vector a onto the coordinate axes. The vector lines (f lines of force, lines of flow) of a vector field are found from the system of differential equations

A scalar or vector field which does not depend on the time t is said to be stationary, if it depends on the time, non-stationary.

2.12.2 Gradient. The vector

where

is the Hamiltonian operator (del or nabla), is called the gradient of the field U=f(P) at the given point P . The gradient is in the direction of the normal n to the level surface at the point P, in the direction of the increasing function U, and has length

.

If the direction is given by the unit vector

,

then

,

(the derivative of the function U in the direction l).

2.12.3 Divergence and rotation. The divergence (div) of a vector field

is the scalar

The rotation (curl) of a vector field

is the vector

2.12.4 Flux of a vector. The flux of a vector field a(P) through a surface in a direction defined by the unit vector of the normal n {cos a, cos b, cos g) to the surface S is the integral

If S is a closed surface bounding a volume V and n is a unit vector of the outer normal to the surface S, then the Ostrogradsky-Gauss formula applies, which has the vector form

Control questions

1. How you can define a scalar field?

2. How you can define a vector field?

3. From what equation you find the vector lines of the field a=f(P)?

4. What is called the gradient of the field U=f(P)?

5. What is the Hamiltonian operator (nabla) of a vector field?

6. What formula describes the derivative of the function U in the direction l?

7. What is called the divergence (div) of a vector field ?

8. What is called rotation (curl) of a vector field ?

9. What is called the flux of a vector field ?

10. What formula is named the Ostrogradsky-Gauss formula?