Vector Fields

Fall 2014, Math 211, Multivariable Calculus, Carleton College, Prof. Joshua R. Davis

2D gravitational potential field (up to sign)

We’ve seen this vector field before now (on our first exam). It’s the acceleration field of a gravity well.

f = (x^2 + y^2)^(-1 / 2)
v = Grad[f, {x, y}]
VectorPlot[v, {x, -1, 1}, {y, -1, 1}, VectorPoints->20]

2D vortex

Here’s another basic example of a 2D vector field. Interpreted as a velocity field, it’s kind of like a whirlpool or vortex.

v = {-y / (x^2 + y^2), x / (x^2 + y^2)}
VectorPlot[v, {x, -1, 1}, {y, -1, 1}, VectorPoints->20]

2D Lotka-Volterra

After examining the vector field plot, make a second plot using the StreamPlot command instead of the VectorPlot command (with all of the arguments to the command identical). Change the values of the parameters a, b, c, d, to examine their effect on the system. Can you change them to make the x-quantity crash? Can you make the x-quantity explode?

a = 1.5;
b = 1.0;
c = 3.0;
d = 1.0;
v = {a x - b x y, d x y - c y}
VectorPlot[v, {x, 0, 10}, {y,0, 10}, VectorPoints->20]

3D gravitational potential field (up to sign)

Copy, paste, and edit the 2D gravitational potential field example from above, to make it 3D. Also, what are the curl and divergence of this 3D vector field? Is this vector field conservative?

3D vortex

Plot the vector field for this 3D vortex. You may want to decrease VectorPoints from 20 to 10. Also, what are the curl and divergence of this vector field? Is this vector field conservative? (This last question is pretty difficult.)

v = {-y / (x^2 + y^2), x / (x^2 + y^2), 0}

Divergence of curl

By experimentation, find a 3D vector field of non-zero curl. Compute its curl. Compute the divergence of its curl.