As previously indicated, many quantities of interest to physicists can be described as vectors having both magnitude and direction. However, some of these quantities can also have a continuous range of values, which then often requires the additional methodology of calculus to be applied. The key vector calculus operations are:
At this point, the mathematical notation required to describe these functions can become a little daunting, but the basic principles previously described still apply. The gradient, divergence and curl of scalar and vector fields are defined by partial differentiation summarised in terms of the [∇] operator, which is really just a shorthand way of writing the rate of change occurring in three dimensions, i.e. x, y, z:
The symbols i, j & k are the unit vectors associated with the scalar magnitude in each direction, i.e. x, y & z, while ∂/∂n is simply the rate of change of magnitude written as a partial derivative. The mathematical concept of gradient, divergence and curl then use this basic [∇] operator to describe different functions. Unfortunately, it is not so easy to come up with an intuitive example against which to describe these concepts. However, it might be useful to introduce the concept of a 'flux' by way of an example within vector calculus, as it will also be needed when we discuss Maxwell's electromagnetic equations in a later section. So, in this context, we shall describe a 'flux' as a measure of a flow, e.g. magnetism, passing through a given surface, which requires the following definitions:
- The source of the flux in terms of strength and direction.
- The shape, size and orientation of the surface
In the context of this definition, we might define a flux in terms of an integral that sums field strength [F] over the total surface area [A]:
The total flux also depends on the size and orientation of the surface integrated. Only when this surface is perpendicular to the vector field lines will it capture the maximum flux per unit area. However, this facet can be expressed in a geometric form of the flux equation:
For now [F] can be considered as a generic vector field, where the angle [θ] exists between the surface and the field lines. As such, the size depends on the orientation and is analogous to the `shadow` of the surface being projected onto a plane perpendicular to the field lines. We will see later how this facet corresponds to the definition of a `dot product` in vector calculus.
The gradient is a vector operation that produces a vector, whose magnitude is the maximum rate of change and points in the direction of the maximum rate of change, e.g. we could define a temperature gradient as:
As such, the gradient is simply the rate of change of a function. It is a vector that:
- Points in the direction of greatest increase of a function
- Is zero at a local maximum or local minimum
A single derivative function, such as dF/dx, tells us how much the field [F] changes for a given change in [x], but a multiple variable field [F], involving x, y & z, will have multiple derivatives, i.e. dF/dx, dF/dy & dF/dz. Thus, the gradient of a multi-variable function has a component for each direction, which collectively point in the direction of the greatest rate of change. The following is just an example function of the field [F] that can be differentiated to find the maximum rate of change:
Therefore, in this 3-dimensional example, we could use Cartesian coordinates to specify an initial values of x, y, z = 2, 3, 4. If we substitute these values into the solution of the gradient derivatives, we get the gradient to be (1, 6, 48), i.e. it is a location in 3-D space that points in the direction of greatest increase of the function for the initial location specified. However, it should be noted that this gradient may only lead to a localise maxima, analogous to a hilltop from which you can see higher mountains.
The divergence is an operator that measures the magnitude of a vector field at a given point; although this might be either a source or a sink point. The divergence of a vector field is a signed scalar, which results from the dot products of two vectors, i.e. Del [∇] and the vector field in question. Remember that the Del [∇] operator is really just a shorthand way of writing the rate of change occurring in three dimensions, i.e. x, y, z, which itself has the vector properties of magnitude and direction. As such, divergence, being a dot product, tells us the magnitude of the flux going in or out of an infinitesimally small volume:
As such, divergence can be thought of as the rate of expansion or contraction at a given point, but as a scalar it has magnitude, but no direction. By way of an example to illustrate the mathematical principle, let:
In this example, F(x,y) is a vector that can be described in two-dimensions, i.e. x & y, where i & j represent the unit directional vectors and the expressions, in brackets, corresponds to the magnitude. Therefore, by substituting into the previous general equation for divergence, reduced to two-dimensions, we get:
So, the divergence is `flux density` or more simply the amount of flux entering or leaving a point. In this specific example above, divergence depends only on the value of [x], so at a point associated with [x=2), divergence has a magnitude of 14.
In mathematical terms, the curl of a vector function is the vector or cross product of the del [∇] operator with another vector function. As such, the result must be another vector that is perpendicular to the input vectors. In this context, a curl is described as a vector operator that shows a vector field's rate of rotation, i.e. both the direction of the axis of rotation and the magnitude of the rotation. It is also sometimes described as the 'circulation density'. So, in the current context, circulation is being caused by a vector field, e.g. a 'force' field, which pushes along a closed boundary or path, e.g. a circle. The Curl operator is describing the circulation per unit area or rate of rotation at a single point. To illustrate by example, a whirlpool clearly has rotation and therefore some force must be the source of that rotation. However, if the whirlpool reduces in size, while the force remains constant, there must be a higher concentration of rotation around the central point, i.e. it has a higher curl. If the whirlpool widens, it would have a smaller curl. As indicated, a curl is the cross product of two vectors, which gives rise to a resultant vector, i.e. the curl, which has both magnitude and direction.
The tabular form on the right is essentially just a shorthand way of writing an expression, which can be expanded using the rules of determinants to give:
Where F(x,y,z) represents the magnitude of the rotational force in three-dimensions at a given point and [∇] represents the rate of change or gradient of the field. As a cross product, the curl must be perpendicular to both input vectors. In respect to our whirlpool example, the rotation when viewed from above is constrained to two-dimensions, i.e. [x,y], the force vector acts along the line of a circular path, while the gradient changes along the radius from a central point. As such, in this specific case, the direction of the curl must be perpendicular to both x-y axes, i.e. it is rotating around the z-axis. Again, let us do another example just to get familiar with the mathematical manipulation:
At this point, we will defer putting too much physical interpretation on the meaning of curl until we come to discuss Maxwell's electromagnetic equations. However, anybody who has played with a magnetic field produced by an electric current passing through a wire will know the magnetic field is both circular and perpendicular to this current.