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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.

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