I'd like to check my understanding of the following integral ( and hopefully, in the process, provide a page where other students can come to understand it as something other than a visual stimulus for beginning a soon-forgotten procedure ):
$\oint_c \vec{F} \cdot \,d\vec{r}$
This line integral is often seen in Stoke's Theorem, Green's Theorem, and many, many classes in multivariable calculus.
In it, we have the following components:
A Vector Field$\ \ \ \ \vec{F}$
- It might be used to model the movement of a liquid or gas. Imagine yourself underwater in the ocean, the current pushing and pulling various points of your body in different directions; this is a vector field.
A Vector-valued Function$\ \ \ \ \vec{r}$
Imagine putting a tube in the water with you; give it a name, call it C. It remains stationary in the water near you, looped around where you float some distance away. It has magical properties: it does not disturb the vector field; it merely exists in it. Water flows freely through its sides, as though its walls weren't even there. It allows fluid to flow through it, but it is infinitesimally small.
If you were to describe where it is in the water relative to you, you might imagine a beam of light emanating from your head, allowing you to describe how far $\vec{r}$ is from your head, and at which angle your head must turn to look at any point on $\vec{r}$. These measurements are scalar, and the inputs required by $\vec{r}$.
The Derivative of $\ \vec{r}$$\ \ \ \ \ \ \,d\vec{r}$
- Given the correct input, $\,d\vec{r}$ will tell you the location of a point in $\vec{r}$ and the direction from that point to the next point in $\vec{r}$
The Dot Product of$\ \ \ \ \ \ \vec{F} \cdot \,d\vec{r}$
- This tells you how much motion water has parallel to $\,d\vec{r}$ at a single point on $\vec{r}$
The Operation$\ \ \ \ \oint_c$
This operator requires a set of instructions as input. For example, "Measure the amount of motion water has at a single point parallel to $\,d\vec{r}$". Then, it says, "Now, take that same measurement at a lot of places on $\vec{r}$, starting here and ending here. Write them all down, sum them all up, and give me the result."
The loop in the middle lets us know that $\vec{r}$ forms a loop--it's a simple, closed path--and we are going to start taking our measurements somewhere on $\vec{r}$ and keep going, all the way around, until we end up right back where we started--thus, taking all our measurements at every single location along $\vec{r}$
So, $\oint_c \vec{F} \cdot \,d\vec{r}$ says:
Measure how much water is flowing parallel to $\,d\vec{r}$ at every single point along $\vec{r}$, sum all those measurements up, and return to me the result.