Why does it seem the inner curls within a surface always cancels in order for greens theorem to be true

Question

Im trying to learn aerodynamics in general for my course. Every video i see to derive the concept of greens and stokes theorem shows how the inner curls within a surface area cancel to 0 and its only the outer surface edge that has a contribution from the vectors. Hopefully the image better shows this…

But what if there was a segment where the curl wasn’t exactly the same as any adjacent to it? Surely the chance of the curl been the same on each segment is practically never seen? Why and how did this assumption come from?

Essentially I don’t see how say there was a vector field stated below, how the outer edge describes the inner surface…

Answer 1

We are not really comparing "adjacent" line segments; all we are saying, more precisely, is that the circulation along a line segment is the same as itself. This is hard to see from a visual aid like the one in the question, because in order to make the diagram legible, we're forced to draw the arrows a bit away from the boundary they correspond to.

However, for example, the two arrows highlighted in red below really both mean "take the circulation along the line segment highlighted in orange".

That segment is the segment where two adjacent "cells" of the division touch. If we take the counterclockwise circulation around the boundaries of both cells, then both circulations will include the circulation along that segment, but in opposite directions. Line integrals along the same segment in opposite directions are negatives of each other, so these two circulations cancel; this does not require any coincidental relationship between the values of the vector field in different places.

Answer 2

Imagine you know that the fundamental theorem of calculus:

$$\int_a^b f'(x) dx = f(b) - f(a)$$

only holds for differentiable functions defined on $[0,1]$. Then how would you show, say, that for differentiable $f : [0,3] \to \mathbf{R}$ the theorem holds?

We can define $f_k(x) =f(x+k)$, translation of $f(x)$ to the left by $1$, and observe that $f_k(x)$ restricted to $[0,1]$ is just $f(x)$ on $[k,k+1]$. But we know that we can use the fundamental theorem of calculus on $f_k(x)$ restricted to $[0,1]$, so that:

$$\int_0^1 f'_k(x) dx = f(k+1) - f(k)$$

Moreover, the integral is the area under the curve, we have that:

\begin{align*}\int_0^3 f'(x)dx &= \int_0^1 f'_2(x)dx + \int_0^1 f'_1(x)dx + \int_0^1 f(x)dx\\ &= f(3) - f(2) + f(2) - f(1) + f(1) - f(0)\end{align*}

Which is just $f(3) - f(0)$! In this way we have used the fundamental theorem of calculus on a "tiling" of $[0,3]$, and observed that because our tiles only overlap at their ends, the sum of the integrals ends up becoming a telescoping sum, and everything except the first and last term cancel. In effect, we have used the fundamental theorem of calculus, which we pretended only works for $[0,1]$, to prove the fundamental theorem for $[0,3]$.

The proof of Green's Theorem is exactly the same, but now it's two-dimensional. We first prove Green's theorem for squares, then tile our region with squares that overlap on their ends. Then just like how the interior ends in the one-dimensional case cancel, the interior boundaries in the two-dimensional case cancel as well. Thus the only thing that matters for Green's theorem is the outer boundary of the region integrated over, which is precisely what we wanted to show.

Answer 3

This can be explained by a small example.

Say you take a wooden plank amd apply an anticlockwise rotational force (like curls given in the picture) at different points. Consider a point P between any two points of application of torque (rotational force), say A & B. The tangential forces experienced at the point P due to torque at A and B are opposite to each other.

The opposite tangential forces cancel out each other in the plank and what the plank will experience is the total rotational force applied as though applied only at the edges where the tangential forces don't cancel out.

This is a very blank simplistic explanation I got.