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. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines?   (b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y.   (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).

Question

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.

(a) Use a sketch of the vector field F(x, y) = xi − yto draw some flow lines. From your sketches, can you guess the equations of the flow lines?
 
(b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y.
 
(c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).

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