Show a flowline of a vector field?

[Suy]

Show a flowline of a vector field??

Homework Statement

Consider the vector field F(x,y,z)=(8y,8x,2z).
Show that r(t)=(e^8t+e^−8t, e^8t−e^−8t, e^2t) is a flowline for the vector field F.

r'(t)=F(r(t)) = (_,_,_)

Now consider the curve r(t)=(cos(8t), sin(8t), 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.

G(x,y,z)=(_,_,_)

Homework Equations

The Attempt at a Solution

I guess the first part of the question r'(t)=F(r(t)) = (8e^8t-8e^-8t,8e^8t-8e^-8t,2e^2t)

For the second part, I don't understand the question... hope someone can explain to me?

 

[HallsofIvy]

Homework Statement

Consider the vector field F(x,y,z)=(8y,8x,2z).
Show that r(t)=(e^8t+e^−8t, e^8t−e^−8t, e^2t) is a flowline for the vector field F.

r'(t)=F(r(t)) = (_,_,_)

Now consider the curve r(t)=(cos(8t), sin(8t), 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.

G(x,y,z)=(_,_,_)

Homework Equations

The Attempt at a Solution

I guess the first part of the question r'(t)=F(r(t)) = (8e^8t-8e^-8t,8e^8t-8e^-8t,2e^2t)

Okay, and do you understand why that is "(8y, 8x, 2z)"?

For the second part, I don't understand the question... hope someone can explain to me?

Do the same thing. If $r(t)= (x, y, z)= (cos(8t), sin(8t), e^{2t})$ what is $r'(t)$? What is that in terms of x, y, and z?

 

[Suy]

Thanks for the reply! It definitely helped me understanding the question! And I know how to do it now!