Posts tagged with vector fields










Late spring. Just after the last frost. I was walking home barefoot, facing the sun. Above one yard with plenty of white dandelion heads I saw in the polarised light golden motes dancing, circling each other, bobbing and teasing each other. They looked like golden fairies.




a smooth field of 1-vectors in 3-D

a smooth field of 1-vectors in 3-D

(Source: thievess)


hi-res




http://i.imgur.com/uUphG.png

(I had to look up MLLW—it means mean lower low water. As in there’s a lower high tide, a higher low tide, a lower low tide, and a higher high tide.) Normal tide cycle from Wikipedia:
http://upload.wikimedia.org/wikipedia/en/0/00/Tide_type.gif

9449880 Friday Harbor, WA water level/meterological data plot

http://upload.wikimedia.org/wikipedia/commons/e/e8/Sloshrun.gif

http://www.hpc.ncep.noaa.gov/sfc/lrgnamsfcwbg.gif

(Source: basecase.org)




A beautiful depiction of a 1-form by Robert Ghrist. You never thought understanding a 1→1-dimensional ODE (or a 1-D vector field) would be so easy!

What his drawing makes obvious, is that images of Phase Space wear a totally different meaning than “up”, “down”, “left”, “right”. In this case up = more; down = less; left = before and right = after. So it’s unhelpful to think about derivative = slope.

BTW, the reason that ƒ must have an odd number of fixed points, follows from the “dissipative” assumption (“infinity repels”). If ƒ (−∞)→+, then the red line enters from the top-left. And if ƒ (+∞)→−∞, then the red line exits toward the bottom-right. So no matter how many wiggles, it must cross an odd number of times. (Rolle’s Thm / intermediate value theorem from undergrad calculus / analysis)

Found this via John D Cook.

(Source: math.upenn.edu)




Vector fields pervade. I think about them every time I throw a frisbee in wind.

image

In a social context, I think about vectors of intent attached to people talking at a party — vectors of flirtation, vectors of eye movement and attention, and more abstract vectors representing jokes, topics of discussion, dance moves, or songs that are playing.

image

Also when I’m thinking about international trade or just the local flows of money in my community, it’s natural to use the vector-field metaphor to “see” the flows.

Electric field of 3 point charges

I also think of history (at different scales) using vector fields. Wars are like nation-states or soldiers aiming weapon vectors at each other. Commerce has many more dimensions since goods and money are both multi-dimensional. Ideas and culture also transmit in a vector-field-like way. Epidemics — well, there’s a reason mosquitoes are referred to as disease vectors.

image

Information flows, thoughts, internet bits — anything that can be characterised as a vector, you can expand that thought into a more complicated vector-field thought. Turbulent versus laminar flows of ideas and culture? Maybe it wouldn’t deserve a research grant but it’s fun to think about.

image

There are pretty obvious physical examples of vector fields — rivers, wind, geological eroding forces, magnetism, gravity, flying machines, bridge engineering, parachute design, weather patterns, your entire body as it does martial arts or dances. Being measurable, these are the source of most of the neat vector-field pictures you can find online.

image

(Or you find programmatically simple theoretical vector fields like the above: a vector facing [−y,x] is attached to every point (x,y). So for instance the point (3,4) has a pointer going out −4 south and 3 east, which equals a total force of 5.)

image

The same metaphors and visualisations, though, are open to interpretation as social or economic variables too. For example a profitable business is more of a “sink” or attractor for 1-D money flows, while a benefactor is a “source”. Likewise a blog that receives lots of links and traffic is a 2-D attractor on the graph of the web — and Google recognises that as PageRank.

image

I know of at least one paper that tries to best economists’ utility theory models by imagining a person on a 1-D vector field, trying to avoid minus signs and find a path to plus signs in the space.

Lotka-Volterra-Goodwin Predator-Prey Model

There is also a game theory connection. Basins of attraction can draw you into a locally optimal place that is not globally optimal. You can imagine examples in the evolution of animals, in company policies or business practices, or in whole economic systems.

image

image

On the one hand it may seem frivolous or crackpottical to generalise these concrete physical concepts to the social or psychological. On the other hand — that’s the power of the generality of mathematics!

image

Vector fields are surfaces or spaces with a vector at each point. That’s the mathematical definition.