This is a continuation of my earlier post on smooth infinitesimal analysis. In this installment, I’ll show how the definition of a “stationary point” in Smooth Infinitesimal Analysis leads directly to a nice substitute for the Lagrange multipliers method. Then I’ll show how you can define differential forms as objects which assign a “signed volume” to genuinely infinitesimal objects, and how you can get Stokes’s Theorem (and the divergence theorem, etc.) in SIA.

**Multivariable Calculus **

*Definition *[Partial Derivatives].

Let be a function from to . We define the partial derivative (also written ) as follows: Given , let . Then is defined to be . A similar definition is made for , and for functions of more than two variables.

*Definition* [].

For , let . Note that .

The sets play the role in multivariable calculus that played in singlevariable calculus. For example, we have the following.

*Proposition.* Let be a function from to . Then, for all ,

and furthermore, and are unique with those properties.

The analogous statement is also true for functions of more than two variables.

We also have

*Proposition* [Extended Microcancellation]. Let . Suppose that for all , . Then each equals 0.

**Stationary Points and Lagrange Multipliers**

There is an interesting substitute for the method of Lagrange multipliers in Smooth Infinitesimal Analysis. To introduce it, I’ll first discuss the concept of stationary points.

Suppose that we’ve forgotten what a stationary point and what a critical point is, and we need to redefine the concept in Smooth Infinitesimal Analysis. How should we do it? We want a stationary point to be such that every local maximum and local minimum is one. A point gives rise to a local maximum of a single-variable function just in case there is some neighborhood of such that for all in that neighborhood.

However, in Smooth Infinitesimal Analysis, there is always a neighborhood of on which is *linear*. That means that for to be a local maximum, it must be \emph{constant} on some neighborhood. Obviously, the same is true if is a local minimum. This suggests that we say that has a stationary point at just in case for all .

*Definition* [Stationary Point of a Single-Variable Function]. Let and . We say that has a stationary point at if for all , .

Similarly, given a function of two variables, and a point , is linear on the set . This suggests that we define to be a stationary point of just in case for all .

*Definition* [Stationary Point of a Multivariable Function]. Let . We say that is a stationary point of if for all , .

Now, suppose we want to maximize or minimize a function subject to the constraint that it be on some level surface , where is a constant. Now, we should require of not that for all , but only for those which keep on the same level surface of ; that is, those for which . I’ll record this in a definition.

*Definition* [Constrained Stationary Point]. Let , . A point is a stationary point of constrained by if for all , if then .

I’ll show how this definition leads immediately to a method of solving constrained extrema problems by doing an example.

This example (and this method) are from [Bell2]. Suppose we want to find the radius and height of the cylindrical can (with top and bottom) of least surface area that holds a volume of cubic centimeters. The surface area is , and we are constrained by the volume, which is .

We want to find those such that for all those such that . So, the first question is to

figure out which satisfy that property.

We have

which is

If this is to equal , then we must have , so that .

Now, we want to find an so that where .

We have

which is

If this is to equal then we must have . Substituting , we get . By microcancellation, we have , from which it follows that .

**Stokes’s Theorem**

It is interesting that not only can the theorems of vector calculus such as Green’s theorem, Stokes’s theorem, and the Divergence theorem be stated and proved in Smooth Infinitesimal Analysis, but, just as in the classical case, they are all special cases of a generalized Stokes’s theorem.

In this section I will state Stokes’s theorem.

*Definition.* Given , , we say that if . We define to be the set .

*Definition.* Let be a curve, and be a vector field. The line integral is defined to be .

*Definition.* Let be a surface, and be a function. The surface integral is defined to be .

This definition may be intuitively justified in the same manner that the arclength of a function was derived in an earlier section.

*Definition.* Let be a surface, and be a vector field. The surface integral is defined to be

Note that this equals .

We extend both definitions to cover formal -linear combinations of curves and surfaces, and we define the boundary of a region to be the formal -linear combination of curves .

The curl of a vector field is defined as usual, and we can prove the usual Stokes’s Theorem:

*Theorem*. Let be a surface and a vector field. Then

This theorem may be used to compute answers to standard multivariable calculus problems requiring Stokes’s theorem in the usual way.

As an exercise, state the divergence theorem in SIA.

**Generalized Stokes’s Theorem**

The definitions in this section are directly from [Moerdijk-Reyes].

*Definition *[Infinitesimal -cubes]. For , and any set, an infinitesimal -cube in is some where and .

Intuitively, an infinitesimal -cube on a set is specified by saying how you want to map into your set, and how far you want to go along each coordinate.

Note that an infinitesimal 0-cube is simply a point.

*Definition *[Infinitesimal -chains]. An infinitesimal -chain is a formal -linear combination of infinitesimal -cubes.

*Definition* [Boundary of -chains]. Let be a 1-cube . The boundary is defined to be the 0-chain , where this is a formal l-linear combination of 0-cubes identified as points.

Let be a 2-cube . The boundary is defined to be the 1-chain .

In general, if is an -cube , the boundary is defined to be the -chain .

The boundary map is extended to chains in the usual way.

*Definition* [Differential Forms]. An -form on a set is a mapping from the infinitesimal -cubes on to satisfying

1. Homogeneity. Let , , and . Define by . Then for all , .

2. Alternation. Let be a permutation of . Then , where and .

3. Degeneracy. If , .

We often write as

We extend to act on all -chains in the usual way.

These axioms intuitively say that is a reasonable way of assigning an oriented size to the infinitesimal -cubes.

The homogeneity condition says that if you double the length of one side of an infinitesimal -cube, you double its size.

The alternation condition says that if you swap the order of two coordinates in an infinitesimal -cube, then you negate its oriented size.

The degeneracy condition says that if any side of the infinitesimal -cube is of length 0, its oriented size is of length 0.

By the Kock-Lawvere axiom, for all differential -forms , there is a unique map such that for all and we have .

*Definition* [Exterior Derivative]. The exterior derivative of a differential -form is an -form defined by

for all infinitesimal -cubes.

*Definition* [Finite -cubes]. A finite -cube in is a map from to .

The boundary of a finite -cube is defined in the same way that the boundary of an infinitesimal -cube was defined.

In the above section, a curve was a finite 1-cube in and a surface was a finite 2-cube in .

*Definition* [Integration of forms over finite cubes]. Let be an -form on and a finite -cube on . Then is defined to be

Generalized Stokes’s theorem (for finite -cubes) is provable in SIA (see [Moerdijk-Reyes] for the proof).

*Theorem* [Generalized Stokes’s Theorem]. Let be a set, an -form on , and a finite -cube on . Then

Let’s see how this gives the Fundamental Theorem of Calculus.

Let and let . We would like to see how is a special case of Generalized Stokes’s Theorem. (On the other hand, that it’s *true* is immediate from the way we defined integration.)

Let be the 0-form on defined by . (Recall that 0-cubes are identified with points.)

Then is the 1-form which takes infinitesimal 1-cubes to . We must show that for the finite 1-cube , .

The boundary of is (as a formal linear combination, not as a subtraction in ). Therefore, . Since is a function from to , there is a unique such that for all . Then . Therefore, , where for all .

Therefore, .

One can show in a similar manner that Stokes’s theorem and the Divergence theorem are special cases of Generalized Stokes’s theorem, although the computations are significantly more arduous.