Question

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

c(t) = (2 sin(t), 2 cos(t), 9et); F(x, y, z) = (y, −x, z)

c'(t) = ?

F(c(t)) = ?

Answer 1

a)
`c'(t) = (2 Cos(t), -2 Sin(t), 9e^t)`

b)
`c'(t) = (2 Cos(t), -2 Sin(t), 9e^t)`

Explanation:

We are given in the question:

`c(t) = (2 Sin(t), 2 Cos(t), 9e^t)`

F(x,y,z) = (y, -x, z)

a)
`c'(t)`

We differentiate with respect to t.

`c'(t) = (2 Cos(t), -2 Sin(t), 9e^t)`

b) F(c(t))

This is a composite function.

`F(c(t)) = F(2 Sin(t), 2 Cos(t), 9e^t)`

`= (2 Cos(t), -2 Sin(t), 9e^t)`