This is a new multivariable blogging site that we'll be updating very often about what was learned in class. If you're ever absent, feel free to visit this site to catch up on what we did. We hope this will help you.

On Friday, we learned about flow lines (stream lines). One of the questions in the Utexas homework assignment basically explains what these are: "The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines."

The path of a particle is given by the position function s(t) = < x(t), y(t) >, which are the parametric equations of the flow lines for a vector field F(x,y). You plug them into the separable differential equation dy/dt / dx/dt (or dy/dx) and find its antiderivative after separating all the "x'es" to one side and all the "y's" to another side to get the equation of the flow lines of the vector field. Don't forget to add the constant (C)!

For example, given F(x, y) = < y, x >, find the equation of its flow lines.
dy/dt = x; dx/dt = y --> dy/dx = x/y (separable differential equation)
After separating the "x'es" and "y's," you get ydy = xdx.
Antiderive each side to get ½ y² = ½ x² + C, which is equivalent to y²/2 - x²/2 = C. This is the equation of the given vector field's flow lines. Can you recognize what kind of graph that is?

Hope this was helpful.