This Quantum World/Appendix/Fields

FieldsEdit

As you will remember, a function is a machine that accepts a number and returns a number. A field is a function that accepts the three coordinates of a point or the four coordinates of a spacetime point and returns a scalar, a vector, or a tensor (either of the spatial variety or of the 4-dimensional spacetime variety).

GradientEdit

Imagine a curve   in 3-dimensional space. If we label the points of this curve by some parameter   then   can be represented by a 3-vector function   We are interested in how much the value of a scalar field   changes as we go from a point   of   to the point   of   By how much   changes will depend on how much the coordinates   of   change, which are themselves functions of   The changes in the coordinates are evidently given by

 

while the change in   is a compound of three changes, one due to the change in   one due to the change in   and one due to the change in  :

 

The first term tells us by how much   changes as we go from   to   the second tells us by how much   changes as we go from   to   and the third tells us by how much   changes as we go from   to  

Shouldn't we add the changes in   that occur as we go first from   to   then from   to   and then from   to  ? Let's calculate.


 


If we take the limit   (as we mean to whenever we use  ), the last term vanishes. Hence we may as well use   in place of   Plugging (*) into (**), we obtain

 

Think of the expression in brackets as the dot product of two vectors:

  • the gradient   of the scalar field   which is a vector field with components  
  • the vector   which is tangent on  

If we think of   as the time at which an object moving along   is at   then the magnitude of   is this object's speed.

  is a differential operator that accepts a function   and returns its gradient  

The gradient of   is another input-output device: pop in   and get the difference

 

The differential operator   is also used in conjunction with the dot and cross products.

CurlEdit

The curl of a vector field   is defined by

 

To see what this definition is good for, let us calculate the integral   over a closed curve   (An integral over a curve is called a line integral, and if the curve is closed it is called a loop integral.) This integral is called the circulation of   along   (or around the surface enclosed by  ). Let's start with the boundary of an infinitesimal rectangle with corners       and  

The contributions from the four sides are, respectively,

  •  
  •  
  •  
  •  

These add up to

 

Let us represent this infinitesimal rectangle of area   (lying in the  -  plane) by a vector   whose magnitude equals   and which is perpendicular to the rectangle. (There are two possible directions. The right-hand rule illustrated on the right indicates how the direction of   is related to the direction of circulation.) This allows us to write (***) as a scalar (product)   Being a scalar, it it is invariant under rotations either of the coordinate axes or of the infinitesimal rectangle. Hence if we cover a surface   with infinitesimal rectangles and add up their circulations, we get  

Observe that the common sides of all neighboring rectangles are integrated over twice in opposite directions. Their contributions cancel out and only the contributions from the boundary   of   survive.

The bottom line:  

This is Stokes' theorem. Note that the left-hand side depends solely on the boundary   of   So, therefore, does the right-hand side. The value of the surface integral of the curl of a vector field depends solely on the values of the vector field at the boundary of the surface integrated over.

If the vector field   is the gradient of a scalar field   and if   is a curve from   to   then

 

The line integral of a gradient thus is the same for all curves having identical end points. If   then   is a loop and   vanishes. By Stokes' theorem it follows that the curl of a gradient vanishes identically:

 

DivergenceEdit

The divergence of a vector field   is defined by

 

To see what this definition is good for, consider an infinitesimal volume element   with sides   Let us calculate the net (outward) flux of a vector field   through the surface of   There are three pairs of opposite sides. The net flux through the surfaces perpendicular to the   axis is

 

It is obvious what the net flux through the remaining surfaces will be. The net flux of   out of   thus equals

 

If we fill up a region   with infinitesimal parallelepipeds and add up their net outward fluxes, we get   Observe that the common sides of all neighboring parallelepipeds are integrated over twice with opposite signs — the flux out of one equals the flux into the other. Hence their contributions cancel out and only the contributions from the surface   of   survive. The bottom line:

 

This is Gauss' law. Note that the left-hand side depends solely on the boundary   of   So, therefore, does the right-hand side. The value of the volume integral of the divergence of a vector field depends solely on the values of the vector field at the boundary of the region integrated over.

If   is a closed surface — and thus the boundary   or a region of space   — then   itself has no boundary (symbolically,  ). Combining Stokes' theorem with Gauss' law we have that

 

The left-hand side is an integral over the boundary of a boundary. But a boundary has no boundary! The boundary of a boundary is zero:   It follows, in particular, that the right-hand side is zero. Thus not only the curl of a gradient but also the divergence of a curl vanishes identically:

 

Some useful identitiesEdit