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).
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.
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:
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 identities