Unveiling the Del Operator: A Key Tool in Vector Calculus

What is the Del operator? What is the meaning of Divergence? and its Physical significance. What is the meaning of Gradient? and its Physical significance. What is the meaning of Curl? and its physical significance.

Del operator:

Del operator is one of the most important and useful mathematical operator.  It gives the space rate of variation of a scalar field (The region in which a scalar quantity is expressed as a continuous function of position, e.g. Temperature distribution in a rod) or vector field (The region in which a vector quantity is expressed as a continuous function of position, e.g. velocity field of a flowing fluid).

Many physical quantities are both variables as well as vector (i.e. they have direction as well as their value changes continuously). Thus to solve physical problems involving such physical quantities, several mathematical operations from the field of vector calculus are needed. Three most important vector calculus operations, which find many applications in physics, are the gradient, the divergence and the curl. Del operator performs all these operations.

It is a vector operator, expression of which is:

Of course, the partial differentiation by themselves have no definite magnitude until we apply them to some function of the coordinates.

Vector Calculus Operations

The gradient

The gradient of a scalar function fi (x,y,z) is defined as:

It is a vector quantity, whose magnitude gives the maximum rate of change of the function at a point and its direction is that in which rate of change of the function is maximum. If S is a surface of a constant value for the function fi (x,y,z) then the gradient on the surface defines a vector that is normal to the surface.

Physical significance of Gradient:

Let us consider a metallic rod the one end of which is hot and the other end is cold. Then heat will flow from the high temperature to low temperature. If temperature at any point (x, y) is represented by a function f(x, y), then grad f gives the maximum rate of change of temperature at point (x, y) and its direction is that in which the rate of change of temperature is maximum thus it shows the direction of heat flow.

Similarly, if a constant potential difference is applied across the ends of a metallic wire, then the end of the wire that is connected to the positive terminal will be at the higher potential in comparison to the other end of the wire. Then the electric field at any point (x, y) will be equal to the negative gradient of electric potential V (x, y) at that point (i.e. the magnitude of electric field vector gives the maximum rate of change of electric potential and it is directed along the maximum rate of decrease in potential). The direction of the current will be opposite to the direction of grad V.

The divergence

The dot (or scalar) product of del operator  and a vector field gives a scalar, known as the divergence of the vector field i.e.,

The physical significance of divergence:

The divergence of an electric field vector E  at a given point is a measure of the electric field lines diverging from that point.

Similarly, the divergence of velocity vector v at a given point of a flowing liquid is a measure of the rate of flow of liquid at that point.

Some important points:

Let us assume a closed surface in a vector field.  There are following three possibilities:

• If the flux entering through the surface is less than the flux coming out through the surface then the div V will be positive (It means there is a source of flux inside the surface).
• If the flux entering through the surface is more than the flux coming out through the surface then the div V will be negative (It means there is a sink of flux inside the surface).
• If the flux entering through the surface is equal to the flux coming out through the surface then the div V will be zero. In this case, the vector field v is called the solenoidal field.

The curl

If a vector field has the circulation or rotation at a point, then the curl of the vector field measures circulation per unit area around that point or rate of rotation. Thus, “the curl of a vector field at a point is a vector having a magnitude equal to the maximum circulation at that point and direction of which is along the perpendicular to the plane of circulation”.

Suppose that the vector field represents the velocity field of a fluid flow (such as water or a gas). Now we place a paddle wheel at any point and if it rotates then the field has a curl at that point and if it doesn`t rotate then there is no curl. It is clear that the paddle will turn only when the fluid exerts a greater force on its one side than the other side. It means that the field is uneven and not symmetric around the point.

Let us consider a vector field F . To measure, how much this field is ‘circulating’ at a point whose position vector is r, we will take an infinitesimal area A around that point and evaluate the integral along the closed boundary of area A that is allowed to shrink to zero. However, since the integral  when area shrinks to zero. Thus, we divide by the area A and define the curl at a point  as:

The physical significance of Curl:

The significance of the curl of a vector field arises in fluid mechanics and in the theory of electromagnetism. In the case of fluid flow, the curl of the velocity field measures the angular velocity of rotation and near the eddy current, it is maximum. Similarly, in electromagnetism where a magnetic field is produced due to the flow of electric current through a conductor, the curl of the magnetic field at any point gives the current density (i.e. current flowing per unit area around that point). If the curl of a vector field is zero then such a field is called an irrotational or conservative field.