Intuition behind the Fundamental Theorem of Calculus, Gauss’s and Stokes’ Theorems.
By: Panos Michelakis
Tags: Stokes, Gauss, Theorem, Intuition, Calculus

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FToC) that we all learned in high-school is the following:

Fundamental Theorem of Calculus

The intuition behind the theorem is pretty straightforward. The integral of a derivative of a function from point a to point b is the sum of all the little “df” changes (we can informally say that the “dx” cancel each other out) along the way. This is actually the analogous of a Telescopic Sum, now expressed in the continuous world (integral) instead of the discrete world (series) where most of us first encountered it:

Telescopic series sum

In the above sum, the “inner” terms constantly cancel each other out after every round of summation and we are finally left with only the “boundary” terms, the first and the last one.

What we will now discuss is how we can use the intuition from the above theorem in order to grasp the idea behind Gauss’s and Stokes’ theorems in a more profound level.

Gauss’s Theorem

Gauss’s Theorem, or The Divergence Theorem as it is also known, says that the outward flux of a vector field across a closed two dimensional surface is equal to the sum of the total divergence of that field over the entire volume that the surface encloses. In mathematical form we write:

Gauss’s Theorem

Admittedly, it seems a little bit daunting when first seen, but as we will now explain, the intuition behind it is actually simple and analogous to the FToC that we previously saw.

As we know, the divergence of a vector field at a specific point is a measure of how much the vector field tends to “spread out”, or diverge, at that point.

Illustration of a vector field with positive divergence

Thus, the triple integral on the left-hand side of the above theorem, adds up all these tendencies of the vector field to disperse over the entire volume, “V”, that the surface , “S”, encloses.

The dot product in the above surface integral on the right-hand side “picks up” only the normal ,with respect to the closed surface, component of the vector field. Based on that, the surface integral on the right hand side computes that total outward flux of the vector field over the surface S.

Outward Flux

The reason why the two above quantities, the outward flux at the boundary and the sum of the divergences at all points in the volume, are equal stems from the same argument that we used to intuitively understand the FToC.

In a manner that resembles a generalized Telescopic Sum, when all of the above divergences add up, there will be a lot of cancellation in the middle of the volume due to opposite divergences and the only portion of the vector field that will “survive” is the one that cannot be cancelled out i.e. the normal portion along the boundary of the surface which is another name for the outward flux.

The general idea that one should keep in mind when dealing with these theorems is the following:

The integral of a derivative (could be a standard derivative, a divergence or a curl as we will see) over a region is equal to the value of the function at the boundary of that region.

We will use this idea to intuitively understand the last theorem of this article, the Stokes’ Theorem.

Stokes’ Theorem

Stokes’ Theorem says that the total curl of a vector field on a three-dimensional surface is equal to the circulation of the field along that surface’s boundary. We write:

Stokes’ Theorem

Again, let’s use some simpler words to describe what we just said. Just like Gauss’s Theorem was concerned with the divergence of a vector field, Stokes’ Theorem is concerned with the curl of a vector field.

The curl of a vector field is a measure of how much the vector field curls around the point in question.

Illustration of a vector field with positive curl

Thus, the surface integral on the left-hand side adds up all these tendencies for the vector field to curl along a specific three-dimensional surface S.

The line integral in the right-hand side has a name. It is called the circulation of the vector field and it is a measure of how much the vector field tends to circulate around an oriented curve C (in this case the boundary of the surface S).

By now, the reason why the circulation and the total curl of a vector field with respect to a surface are equal should be obvious. The “opposite” tendencies to curl inside the surface cancel each other out and what is finally left is the circulation of the field, i.e. the curl of the field at the boundary of the surface.

Stokes’ Theorem

Stay curious.