What you always wanted to know about

VECTOR FIELDS

but were afraid to ask

Vector fields! It sounds terribly complicated. I will show you how simple they are.

When you hear that the temperature in New York is 72 degrees, does it mean that it is 72 degrees everywhere in New York? Of course, not. It means that at the moment of the latest recording the temperature at the location of the recording was 72 degrees. In reality the temperature is different at every location in that vast city and it changes also as time goes. Temperature is in fact a function of location (space) and time. As temperature is a scalar quantity, we may say that it is a scalar field. In general, a scalar field is simply a scalar quantity that is a function of space and time. The difference between a scalar and a scalar field is that the former is one single value of the latter. The scalar field exists in all points of space and at any moment of time while the scalar is its value at a certain location at a certain time.

Now think about a vector quantity, for example the flow vector that determines the direction of water flow at a certain point in a river. We can assign such a vector to any point in the river at any moment of time. Thus we arrive at a vector field that can be graphically represented by field lines that are tangential to the direction of the vector at each point. The density of the field lines is proportional to the magnitude of the vector at the particular location and time (Figure 1)..Therefore, a vector field is simply a vector quantity that is a function of space and time. The difference between a vector and a vector field is that the former is one single vector while the latter is a distribution of vectors in space and time. The vector field exists in all points of space and at any moment of time.

As vector fields exist at all points of space, they can be specified along curves and surfaces as well. This is especially important because all laws of electricity and magnetism can be formulated through the behavior of vector fields along curves and surfaces.

Let us start with curves. An arbitrary curve can be uniquely defined by the dl vector field along the curve. The magnitude of this vector is an infinitely small length element of the curve at a certain location and the direction of it is tangential to the curve at the same location. (Please note that on this page vectors are denoted by bold letters in the text but they have arrows above them in the formulae and figures. It is very important to distinguish between vectors and scalars!). As our arbitrary vector field V also exists at all points of the curve (Figure 2), we can form the dot product of the two vectors that is equal to the tangential

component of V multiplied by the magnitude of dl (remember the geometrical meaning of the dot product):

The integral of this quantity along the entire length of the curve is called the circulation of the vector field V along the curve:

(

where L is the length of the curve and the average tangential component of V along the curve is defined as

To define a circulation we need two vector fields: V and dl. The first is the one we want to investigate, the second defines the curve. The circulation of the arbitrary vector field V along an arbitrary curve is the average tangential component of V along the curve multiplied by the length of the curve. (The name "circulation" suggests a closed curve but the definition applies to any curve, open or closed.)

Let us now consider surfaces. An arbitrary surface can be uniquely defined by the dS vector field on the surface. The magnitude of this vector is an infinitely small surface area element at a certain location and the direction of it is perpendicular (normal)l to the surface at the same location.. As our arbitrary vector field V also exists at all points of the surface (Figure 3), we can form the dot product of the two vectors that is equal

to the normal component of V multiplied by the magnitude of dS:

The integral of this quantity over the entire surface is called the flux of the vector field V through the surface:

where S is the area of the surface and the average normal component of V over the surface is defined as

To define a flux we need two vector fields: V and dS. The first is the one we want to investigate, the second defines the surface. The flux of the arbitrary vector field V through an arbitrary surface is the average normal component of V over the surface multiplied by the area of the surface.

The knowledge of the flux and the circulation is very important because all the laws of electricity and magnetism can be defined through these two simple mathematical concepts (see Maxwell's equations).