(896, #25) 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. Sketch some flow lines for the vector field . From your sketches, can you guess the equations of the flow lines. Find an equation of the flow line that passes through the point (1,1) by solving the differential equations dx/dt = x and dy/dt = -y.

Solution:

a) This is the vector field for F.

b) Solving for x and y in terms of t.

A and B are both constants. When we multiply both equations together we get:

When the line passes through the point ( 1, 1 ) we get:

Therefore the equation of the flowline through the point ( 1,1 ) is:

, x>0

Displaying this curve against the flowlines, we see yhat vectors of the field are tangent to the curve:

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by:ch