Question

Show that the given curve c(t) is a flow line of the given velocity vector field F(x, y, z).

Answer 1

Answer:

The given curve c(t) is a is a flow line of given velocity vector field F(x, y, z).

Step-by-step explanation:

We are given the following information in the question:

$c(t) = (t^2, 2t-6, 3\sqrt{t}), t > 0\\\\ F(x, y, z) =(y+6, 2, \frac{9}{2z} )$

Now, we evaluate the following:

$c'(t) = \frac{d(c(t))}{dt} = (2t, 2, \frac{3}{2\sqrt{t}} )$

Now, we have to evaluate:

$F(c(t)) = (2t-6+6, 2, \frac{9}{6\sqrt{t}} ) = (2t, 2, \frac{3}{2\sqrt{t}} )$

When F(c(t)) = c'(t), then c(t) is a flow line of given velocity vector field F(x, y, z).

Since, $F(c(t)) = c'(t)$, we can say that c(t) is a flow line of given velocity vector field F(x, y, z).