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.

See attached file for full problem description.

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.

Consider the vector field F(x,y,z) = 〈5y, 5x, −2z〉.

Show that r(t) = 〈e5t + e−5t, e5t − e−5t, e−2t〉 is a flowline for the vector field F.

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That is, verify that (I trust you here)

r′(t) = F(r(t)) =

Now consider the curve r(t) = 〈cos(5t), sin(5t),e−2t〉. It is not a flowline of the vector field F, but of a vector field G which differs in definition from F only slightly.

only slightly.

.G(x,y,z) = 〈 , , 〉.