We explore the relationship between the gradient, the curl, and the divergence of a vector field.

### A two-dimensional dream

So far we have two fundamental theorems of calculus for functions of two variables.

- The fundamental theorem of line integrals:
- Green’s theorem:

Wouldn’t it be cool if we could *combine* these theorems to make something like: A
double integral, equaling a single integral, equaling a difference! It’s an amazing idea,
and while it is true, we shouldn’t get too excited. In fact each expression above is
equal to zero. Let’s see why.

First note that both sides of the equation: are zero. This is because, with a closed curve, , and so . But this means that Note, this is totally independent of the region of the region and the surface . The upshot: for all functions with continuous second derivatives!

Of course, you could poo-poo the work above and say “I already knew that! This is
just the Clairaut gradient test!” Well, sure, but what we just presented is
*another* reason that . Moreover, this line of reasoning will lead us to new ideas
too. Read-on young mathematician, there is not much further to go in this
course.

### A three-dimensional dream

Working in a similar way to how we worked above, let us recall the fundamental theorems of calculus for functions of three variables.

- The fundamental theorem of line integrals:
- Stokes’ Theorem:
- The Divergence Theorem:

Putting Stokes’ Theorem and the Divergence Theorem together, we find the beautiful
expression: Again, what a fantastic idea, a triple integral, equaling a double integral,
equaling a single integral! However, there is again a rub, the boundary of a closed
curve is *empty*. A closed curve has **no boundary**! Hence: This fact, in turn means
that all of the integrals above, including: Are all equal to to zero! We thus conclude:

- tells us that the circulation along
**any**closed surface is zero, and - tells us that for any vector field , .

Let’s see what you can do now.

### The shape of things to come

Recalling that is the set of differentiable functions from to where **all** of the
derivatives are continuous, we can make the following “chain” of derivatives:

Since the Fundamental Theorem of Line Integrals (FTLI) is in some sense about “undoing” the gradient and Green’s Theorem is in some sense about “undoing” the scalar curl, we can also place these in the diagram:

Working in a similar way, we can make a chain of derivatives in three-dimensions as well:

Since Stokes’ Theorem is in some sense about “undoing” the curl and the Divergence Theorem is in some sense about “undoing” the divergence, we can place these theorems into the diagram as well:

Now let’s get crazy. Let’s put all of our diagrams together into one big diagram:

Checkout the values for in each of the :

*infinitely!*The details of this are beyond the scope of this course. Math folks who know this eventually start calling all of these fundamental theorems “Stokes’ Theorem.”

Alas, our course has come to an end. However, it is really much more of a **beginning,**
as there is **so much more to know.**