Lecture ¹15

# Lecture ¹15. Vector interpretation Stoks’s formula. Potential field; vector line and vector pipes; solenoid field; “nabla” operator

Exercises for Lecture #15

The purpose of work:

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

Theoretical information and examples

Content

2.12.5 Circulation of a vector, the work of a field

2.12.6 Potential and solenoidal fields

2.12.5 Circulation of a vector, the work of a field. The line integral of the vector a along the curve C is defined by

and represents the work done by the field a along the curve C (as is the projection of the vector a onto the tangent to C).

If C is closed, then the line integral (1) is called the circulation of the vector field a around the contour C.

If the closed curve C bounds a two-sided surface S, then Stokes' formula applies, which has the vector form is

where n is the vector of the normal to the surface S; the direction of the vector should be chosen so that for an observer looking in the direction of n the circulation of the contour C should be counter-clockwise in a right-handed coordinate system.

2.12.6 Potential and solenoidal fields. The vector field a(r) is called a potential if

where U = f(r) is a scalar function (the potential of the field). For the potential property of a field a, given in a simply-connected domain, it is necessary and sufficient that it be non-rotational, i.e., rot a = 0. Then, there exists a potential U defined by

If the potential U is a single-valued function, then

in particular, the circulation of the vector a is equal to zero:

A vector field a(r) is called solenoidal if at each point of the field div a = 0; then, the flux of the vector through any closed surface is zero.

If the field is at the same time potential and solenoidal, then

and the potential function U is harmonic, i.e., it satisfies Laplace's equation

where

is the Laplace operator.

Example. Prove that field is potential and restore the potential.

Solution. Let’s mark . Then condition of potential take the type:

, , .

we mark the potential as F and find it from conditions

Control questions

1. What is called the circulation of the vector field a around the contour C?

2. What is the Stokes' formula?

3. What vector field a(r) is called a potential?

4. What vector field a(r) is called a solenoidal?

5. What potential function U is called a harmonic function?

6. What is called the circulation of the vector field a around the contour C?