Mathematical Physics Vol 1

4.2 Vector field

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4.2 Vector field

4.2.1 Vector function. Vector field Let us assign to each point M , of a region D , according to some law, a value of a vector v . We then say that a vector function v = v ( M ) is defined. The set D , to which values of the argument belong, is called the domain or the vector field of this function. Thus the vector field is the domain of a vector function. However, as each point M is determined by a position vector r , the previous definition can be reformulated as follows if to each value of position vector r ∈ V , where V is the three-dimensional vector space with base vectors e i ( i = 1 , 2 , 3), a value of a vector v is assigned according to some law, we say that a vector function of the argument r is defined

v = v ( r )= v 1 e 1 + v 2 e 2 + v 3 e 3 .

(4.33)

We have expressed here the position vector by its components, in relation to a coordinate system, and consequently also the function itself. As in the Cartesian coordinate system

v 1 = v 1 ( x , y , z ) v 2 = v 2 ( x , y , z ) v 3 = v 3 ( x , y , z )

(4.34)

it follows that each of these relations defines a scalar field. Thus, the study of vector fields comes down to the study of three (if observed in a three-dimensional space) scalar fields. Examples include: the gravitational field of the Earth, the velocity field of a moving fluid, the gradient of a scalar function etc. As an example, we can take any analytically given vector function, such as v = xy i − 2 yz j + x k , whose domain is a vector field. Some typical examples of vector fields are represented in figures 4.7 a,b,c,d.

Figure 4.7: Examples of vector fields.

If the vector field is independent of time, it is called a stationary vector field.

Definition A vector line (line of forces) ℓ is the geometric place of the points of a vector field in which the vector function v ( M ) has the direction of the tangent to this line at given points, i.e. a line where at each point the direction of the vector coincides with the direction of the tangent of the curve at that point.