Question

Define a vector field in the plane and in space. Give some physical examples of vector fields.

Short answer

A vector field in a plane is defined as \( F(x, y) = P(x, y)i + Q(x, y)j \) and in space as \( F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k \), where (P,Q,R) are scalar functions of the variables. Examples include electric, magnetic and gravitational fields in physics, as well as velocity vector fields in fluid dynamics, and wind patterns.

Step-by-step solution

Defining a Vector Field in a Plane

The definition of vector field in a plane can be given by the formula \( F(x, y) = P(x, y)i + Q(x, y)j \) where (P,Q) are scalar functions of (x, y) which are the coordinates of the vectors. Each vector points in the direction that the (x, y) value indicates, and its length is determined by the magnitude of the vector.

Defining a Vector Field in Space

The definition of a vector field in space needs three-dimensional visualization. It can be given by the formula \( F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k \) where (P,Q,R) are scalar functions of (x, y, z). The principle is the same as in the plane, but now the vectors can also move along the z-axis.

Physical Examples of Vector Fields

In physics, vector fields are used to model quantities such as electric, magnetic or gravitational fields that associate a vector to every point in space. In the case of electric fields, the vector at a given point defines the direction in which a positive charge would move if it were placed at that point. In fluid dynamics, another instance is velocity vector fields which describe the speed and direction of a moving fluid throughout space. Wind patterns can also be described as a vector field.