Rectangular functions of two variables takes an x-coordinate as an input and outputs a y-coordinate (or vice versa). Three variable rectangular functions take a point in the plane (both x and y) and outputs a point in space (associating a z-coordinate). Vector valued functions take x-values and output vectors.
Finally, vector fields take in points (both x and y), and output vectors.
Keep in mind that vector fields cover the entire domain of the plane, or space, depending on how many dimensions you are working with. In class, we graphed and analyzed some basic vector fields, associating them with some natural phenomena.
1.
The most basic one: v (x,y) = <x , y> is described as consisting of vectors pointing radially outwards with magnitudes proportional to its distance from the origin. We associated it with the "Big Bang," as the universe's expansion is supposedly accelerating.
2.
The reverse of this vector field: v (x,y) = <-x , -y> essentially reverses every vector in the field. We described the resulting field as the opposite of the Big Bang, or the "Big Crunch."
3.
Our third basic vector field looks a little more natural. We associated the field described by: v (x,y) = <y , x> with two water currents colliding with each other. Note that the reverse vector: v (x,y) = <-y , -x> would just have the "entrance" and "exit" of the vectors reversed.
4.
And finally, our favorite of the day, the "Toilet Flush": v (x,y) = <y , -x>. The reverse of this function: v (x,y) = <-y , x> would simply be a toilet being flushed on the other side of the world.
The drawings I added to go with the vector fields aren't entirely correct, but it's just there to give a less cluttered visualization.
To draw your own vector fields, just plot a few vectors at various points in the plane and try to see a pattern. They're usually symmetric, whether it's about the origin in the case of the "toilet flush" or about the lines y = x and y = -x in the case of "water currents".
Happy...vector field-ing?
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