SUNY Geneseo Department of Mathematics
Tuesday, April 25
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
Tomorrow is GREAT Day.
We don’t have class!
Go to talks, performances, posters, etc. instead.
Based on “Flux” and “Circulation” in section 5.2 of the textbook.
Definition: flux is amount of flow (e.g., of a fluid, of energy, etc.) through a surface. (“Flow” is represented by a vector field in which the vectors are, for example, velocities of particles or droplets of the flowing material.)
A flux integral is the total component of a vector field perpendicular to the surface (analogous to how a regular line integral is the total component of the field parallel to a path or surface).
The equation for flux (see the textbook or the examples below for this equation).
Circulation notation and definition.
Notice that circulation integrals are similar the ones we’ve seen already, except restricted to closed paths.
Calculating work as an application of circulation.
What exactly is the definition of circulation?
It’s not as intuitive as flux, but it’s an attempt to measure how much a flow, represented by a vector field, tends to circulate around a surface.
When someone homebrews beer, they need to circulate hot water through a bag of grains in order to give the beer its flavor. Typically this involves swirling the bag around in a kettle of water, but in terms of relative motion that’s the same as imagining a circular flow of water in the kettle and a stationary bag:
To get flavorful beer, you want to good flow of water through the grains. So let’s imagine a simple set-up, modeled as 2-dimensional, and work out the flux of water through the bag. Assume the flow of water around the kettle is given by the simple rotational vector field V(x, y) = ⟨-y, x⟩, and that the bag is a circle at the edge of the kettle.
To find flux, evaluate the flux integral, i.e., the integral of V(r(t)) dotted with n(t), a normal to r(t). Find n(t) by swapping the components of the derivative of r and negating, as in the text. Note that by negating the second component after swapping, you get a vector that points to the right as you move along r(t) in the direction of increasing t — in our case, that means pointing out from the bag of grain. This will be helpful knowledge when interpreting the sign of the flux. Integrate from 0 to 2π, to capture flux across the entire bag.
Strangely, the flux came out to be 0, suggesting that no water flows through the grains. But another possibility is that exactly as much water flows in as flows out. To check this, let’s find the flux across just the side of the bag that water flows in through, or just the side water flows out through. The basic integral is the same as before, only its bounds have changed. (The bounds should be π to 2π if you want to integrate over the side water flows in through, or 0 to π for the side water flows out through. Most people in class just 0 to π, so that’s what’s shown here):
This time we have a positive flux. Recalling that flux is total flow in the direction of the normal to the curve, and that our normal points out of the bag, positive flux corresponds to flow out of the grains, like we expected.
A circulation example.
“Conservative” or “gradient” vector fields — vector fields with some particularly nice properties with respect to physical applications and integration.
Please read “Gradient Fields (Conservative Fields)” in section 5.1 of the textbook.