Differentials, Exact and Inexact

In the calculus we use the concept of the differential to denote increments that approach the infinitesimal. If we have some smooth, continuous function that translates a number or numbers from some domain onto a number or numbers on some range, then the differential of that function consists of the minuscule difference the function exhibits on its range, df(x), due to some minuscule difference in the numbers on the domain, dx. In the case in which the domain and range consist of the set of the real numbers we have the statement that

(Eq’n 1)

which defines the differential of the function.

We note that the exact/inexact distinction between
differentials does not apply to functions of a single variable, because we plot
the domain of the variable on a single one-dimensional entity, the real number
line. But the exact/inexact distinction __does__ apply to functions of more
than one independent variable. We plot those variables on a multidimensional
realm analogous to the real number line, so when we integrate a function on that
realm we can increment the independent variables in ways that trace different
paths between point A and point B.

That fact gives us two possibilities: either the value of the integral does not depend upon the specific path followed or it does have different values for different paths between the same two points. Because two alternate paths between A and B form a closed loop, we can represent those possibilities with the integrals

(Eq’ns 2)

In the first case dφ represents an exact differential; in the second case dφ represents an inexact differential.

If we have a function F such that

(Eq’n 3)

does not depend upon the path we follow in the integration, the F represents a conserved quantity. If that integral truly does not depend upon the path of integration, then it must give us F as a unique function of the variables. That function must have a unique differential with respect to the variables;

(Eq’n 4)

But if we have only the components P=∂F/∂x, Q=∂F/∂y, and R=∂F/∂z, how can we know that Pdx+Qdy+Rdz represents an exact differential?

We apply Leonhard Euler’s test. If we have a differential in the form dφ=Pdx+Qdy+Rdz, then it is exact if the curl of the vector [P, Q, R] equals zero. That curl equals zero if we have

(Eq’ns 5)

That works if and only if P=∂F/∂x, Q=∂F/∂y, and R=∂F/∂z because for "nice" functions (and we always use nice functions in physics) the mixed second derivatives equal each other (the operation of differentiation commutes with itself);

(Eq’n 6)

That equality makes the curl of the vector equal zero. In each of Equations 5 subtract the right side from the left side: we get the components of the curl of [P, Q, R] equated to zero. That statement necessitates that [P, Q, R]=Lφ, as shown above. The dot product of that gradient with an element of displacement gives us the exact differential.

Exact differentials have applications to the theory of forcefields. Inexact differentials appear frequently in classical thermodynamics.

Field Theory

In physics the term field theory generally refers to those mathematical structures that use the nabla, or its tensor counterparts, as the fundamental operator in the theory. The nabla is a set of three differentiation operators that has the form of a vector:

(Eq’n 7)

in which the boldfaced i,j,k represent the unit vectors pointing in the x-, y-, and z-directions.

We start with a scalar field, which we represent with the Greek letter phi. In a scalar field a single value of the field exists at each and every point in the space occupied by the field. The only application of the nabla to a scalar field gives us the gradient of the field, a vector field in which, at every point, the gradient gives us the vector that tells us the magnitude and direction of the greatest rate of change in the scalar field. The equation

(Eq’n 8)

completely specifies the field φ(x,y,z) if we know the value of the field intensity at some point P. Thus at any point Q we can calculate the field intensity as

(Eq’n 9)

Because we know that L×F=0, we know that F represents a conservative field and, equivalently, that FAdl represents an exact differential.

Now we come to Helmholtz’s theorem, the fundamental theorem of vector calculus, which Hermann Ludwig Ferdinand von Helmholtz (1821 Aug 31 – 1894 Sep 08) discovered. Helmholtz’s theorem states that for any vectorfield F satisfying the conditions

(Eq’ns 10)

we can write a description of the vectorfield itself as the sum of an irrotational part and a solenoidal part,

(Eq’n 11)

in which we have

(Eq’ns 12)

In those equations the subscripted ar represents the location of the source, which we represent by the divergence of F and the curl of F respectively.

We can look at that presentation a little more closely. If we have, in three-dimensional space, a vectorfield F whose algebraic description consists of three smooth, continuous functions of the coordinates that fade toward zero as the coordinates tend toward infinity, then each component has three partial derivatives, so the vectorfield itself has nine partial derivatives associated with it. Three of those partial derivatives compose the divergence of the vectorfield,

(Eq’n 13)

The other six partial derivatives come together in pairs as the curl of the vectorfield,

(Eq’n 14)

Helmholtz’s theorem then tells us that we must have as true to mathematics

(Eq’n 15)

in which G represents the Newtonian potential operator (see Appendix II). This equation corresponds to Equation 11, with

(Eq’n 16)

and

(Eq’n 17)

in which φ and A represent the scalar potential and the vector potential of the vectorfield respectively.

Thus we see that in vector calculus we have only two operations with the nabla that yield a vector output – gradient and curl. The nabla represents what we might call the differentiation vector. That fact lets us infer that every vectorfield has associated with it functions ö and A such that Equation 11 stands true to mathematics. The nature of vector differentiation thus necessitates that we have as also true to mathematics the statements that L×F=0 implies that F=Lφ and LAF=0 implies that F=L×A.

Helmholtz’s theorem also requires that a forcefield
emanate from sources. Thus we have LALφ=L^{2}φ=Qδ(r)
and L×(L×J)=L^{2}J=Iδ(r).
If L×F=0,
then LAF≠0
if F≠0
and we have FAdr=dφ.

(Ea’n 18)

We call this equation the Poisson-Laplace equation: Poisson’s equation if ρ≠0 and Laplace’s equation if ρ=0. We also have

(Eq’n 19)

in which we have used the fact that LAA=0. Now how do we know that last statement stands true to mathematics? We know that A comes from the curl of some function (Equation 17) and we know that the divergence of a curl always equals zero, therefore we know that the divergence of A equals zero.

Classical Thermodynamics

We have the master equation dE=ðQ+ðW. The differential dE must represent an
exact differential because E represents the total energy of a system, a
conserved quantity. But work gives us an inexact differential (represented by
the barred lower-case dee) because in a gas, for example, heat put into or taken
out of the gas changes the pressure and thus ensures that expanding or
compressing the gas between V_{1} and V_{2} does not give a
single answer for the work done. Adding ðQ makes the sum exact, but this gives
us more of a condition on ðQ.

Assume that we have a description of our minuscule element of heat as the sum of two components,

(Eq’n 20)

Solving the equation ðQ=0 gives us the condition on the changes df and dg when no heat moves into or out of the system. We get

(Eq’n 21)

The right side of that equation yields a known function of f and g, so the equation gives us a gradient at each point in the f-g plane. We denote the set of curves that follow that set of gradients with S(f,g)=c, in which c represents an arbitrary constant. Note that each of those curves represents a situation in which the system neither loses nor gains heat (ðQ=0).

Now we have as true to mathematics

(Eq’n 22)

If we simply multiply that equation by df we can see that dS properly represents an exact differential. We can solve that equation easily and obtain

(Eq’n 23)

in which T=T(f,g). We know that the third term belongs in that equation because it must certainly stand proportional to F and G separately and yet include some other factor as well. We can also rewrite that equation as two equations,

(Eq’ns 24)

If we substitute that result into Equation 20, we get

(Eq’n 25)

which corresponds to

(Eq’n 26)

Thus we convert the inexact differential of ðQ into the exact differential of dS by multiplying it by the integrating factor 1/T. Thus heat, ðQ, becomes a combination of entropy (dS, an exact differential) and temperature (T, the integrating factor).

Starting with a minuscule amount of heat, represented by an inexact differential, we have produced S as the integral of an exact differential. We identify S as representing a state-of-the-system function and call it entropy. We identify the integrating factor T with the system’s absolute temperature.

We can apply the mathematical analysis above to any inexact differential that involves two independent variables and find an integrating factor that will convert it into an exact differential. However, we may not have the same success with inexact differentials that involve three or more independent variables.

On a final note, we have the master equation of thermodynamics in the usual form

(Eq’n 27)

in which μ represents the chemical potential of the system and N represents the number of molecules with that potential (and technically the chemical potential term should appear as a sum of different chemical species). In this case T, p, and μ appear to represent the components of a kind of vector. But the corresponding coordinates (S, V, and N) do not conform to any kind of rotation transformation, so Equation 27 does not give us the kind of vector relation that we saw represented in Equation 4.

Appendix I: Vector Integrals

If we have a smooth and continuous scalar field, which we represent with ö, then we have as true to mathematics the statement that

(Eq’n A-1)

in which a and b represent the endpoints of the line L over which we carry out the integration. We call this the gradient theorem and it only works properly if we have an exact differential under the integral.

Green’s theorem gives us a special case of Stokes’ theorem, one in which we have a vectorfield F=[M, N] constrained to occupy a plane. In that case we integrate the scalar curl of the field over an area on the plane and correlate the result with that of integrating the field itself over the curve bounding the area:

(Eq’n A-2)

More generally we have Stokes’ theorem, which applies over any surface in three-dimensional space. Expressed as the integral of a curl of F over some portion of the surface, it tells us that

(Eq’n A-3)

Again, we relate the integral of the curl over a patch on the surface to the integral of the vectorfield itself around the boundary of the surface patch.

Finally, we have the divergence theorem, also known as Gauss’s theorem. In this case we have the integral of the divergence of the vectorfield over some volume equal to the integral of the vectorfield itself over the surface bounding the volume. We have, more technically,

(Eq’n A-4)

in which the vector n represents the unit vector normal to the surface at the point of integration.

This seems a little too magical. We find that the integral of some derivative of a function over an extended realm equals the integral of the function itself on the boundary of the realm. It looks like the mathematical analogue of action at a distance; we feel the need to understand how a function (or its derivative) in the interior of a region can determine how the function adds to the integral over the rather arbitrary boundary of the region. One fact seems clear, though: in these equations we have only exact differentials.

Appendix II: The Newtonian Potential Operator

We have the inverse of the negative Laplacian operator in
the Newtonian potential operator. When we have the mathematical description of a
field in Euclidean space we have L^{2}φ=f,
which usually involves the Dirac delta. We then define the Newtonian potential
operator through the statement that

(Eq’n B-1)

in which the star denotes the operation of convolution.

Convolution denotes an operation that melds two functions and yields a third function that we usually identify as modified version of one of the functions. In field theory whenever we have a linear system subject to a superposition principle, a convolution operation appears. We define convolution as an integral transform operating on the product of two functions in which we have reversed and shifted the coordinate of one of the functions;

(Eq’n B-2)

The convolution exists if f(x) and g(x) represent Lebesgue integrable functions and their product is also integrable. We also need f(x) and g(x) to decay rapidly toward zero as the coordinate tends toward infinity. Convolution has the algebraic properties of commutativity

fàg=gàf,

associativity

fà(gàh)=(fàg)àh,

distributivity

fà(g+h)=(fàg)+(fàh),

associativity with scalar multiplication

a(fàg)=(af)àg=fà(ag),

and multiplicative identity

fàδ=f,

in which δ represents the Dirac delta. If ∂ represents any differentiation or difference operator, then we also have

∂(fàg)=(∂f)àg=fà(∂g).

If T_{x} represents a translation operator in accordance with

T_{x}f(X)=f(X-x),

then we also have as true to mathematics the statement that

T_{x}(fàg)=(T_{x}f)àg=fà(T_{x}g).

So when we convolve the Newtonian kernel with the negative Laplacian of the potential we simply get back the potential, as shown in Equation B-1. In the case of a single particle the negative Laplacian equals the electric charge on the particle multiplied by the Dirac delta (to convert it to a density) and divided by the electric permittivity of vacuum. Because of the multiplicative identity property of convolution we then have the Newtonian kernel in D dimensions as

(Eq’n B-3)

in one dimension,

(Eq’n B-4)

in two dimensions, and

(Eq’n B-5)

in more than two dimensions. In that equation V_{D} represents the
volume enclosed within the unit D-sphere in D-dimensional space,

(Eq’n B-6)

in which R=1 (for the unit sphere) and

(Eq’n B-7)

for D=2k (with k representing any natural number) and

(Eq’n B-8)

for D=2k+1. In three dimensions (k=1) we have

(Eq’n B-9)

and, thus,

(Eq’n B-10)

Convolving that with the negative Laplacian of the electrostatic potential gives us

(Eq’n B-11)

Thus we obtain the standard form of the Coulomb potential.

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