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.

Keeping tabs on storm surges at tidesonline.noaa.gov and not liking what I’m seeing. That air pressure graph….

— Charlie Loyd (@vruba) October 29, 2012

(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:

For comparison, a gauge not experiencing a major storm right now: http://tidesonline.noaa.gov/plotcomp.shtml?station_info=9449880+Friday+Harbor,+WA#

— Charlie Loyd (@vruba) October 29, 2012

The GIF near the end of en.wikipedia.org/wiki/Storm_sur… shows a modeled storm surge dramatically.

— Charlie Loyd (@vruba) October 29, 2012

The number of isobar lines on hpc.ncep.noaa.gov/sfc/lrgnamsfcw… should give some sense of how unusually low that is.

— Charlie Loyd (@vruba) October 29, 2012

It’s dated 1500Z = 11:00 a.m. EDT.

— Charlie Loyd (@vruba) October 29, 2012

Want to see what a hurricane looks like on a seismometer? You’re seeing the ocean wave energy traveling to PA. twitter.com/volcanoclast/s…

— Ian Saginor (@volcanoclast) October 30, 2012

(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**.

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.

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.

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.

**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.

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.

(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**.)

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.

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.

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.

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!

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