Vector Fields

In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane . Vector fields are often used to model the speed and direction of a moving fluid throughout space, for example, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.

Fig 1

The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and, under this interpretation, conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow).

Gradient field: Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇). A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that: 

$\displaystyle{V = \nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial x_3}, \dots ,\frac{\partial f}{\partial x_n}\right)}$

The associated flow is called the gradient flow.

Examples

• A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point.
• A gravitational field generated by any massive object is a vector field. For example, the gravitational field vectors for a spherically symmetric body would all point towards the sphere's center, with the magnitude of the vectors reducing as radial distance from the body increases.
• Magnetic field lines can be revealed using small iron filings.
• In the case of the velocity field of a moving fluid, a velocity vector is associated to each point in the fluid.