Find all solutions (integral surfaces)

[mathgirl1]

Find all solutions u(x,y,z) to \(\displaystyle V \cdot \triangledown u = 0\) where V=(1,1,z) and u(r(t)) = constant where r(t)=<x=t, y=0, z=sin(t)>. What are the constants?

It has been a really long time since I've done Diff Eq and just trying to prepare to take a grad level course in the Spring. From the following you will be able to tell that I am really lost. But here is what I've tried.

From this I get that u(x,y,z) = <x, 0, sin x> so that u(r(t)) = <t, 0, sin(t)>
So \(\displaystyle V \cdot \triangledown u = <1,1,z> \cdot <\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}>=<1,1,z> \cdot <1, 0, 0>=<1, 0, 0>\) Not really sure where to go from here and I don't really think the "plug n chug" method for this equation is the correct method to solve.

I found something that \(\displaystyle \frac{dx}{dt}=P(x,y,z), \frac{dy}{dt}=Q(x,y,z), \frac{dz}{dt}=R(x,y,z)\) So then I would have 1=1 0=1, cos(t)=z? Which doesn't even make sense to me. But I figure I have my u(x,y,z) wrong?

Obviously I am at a total loss and haven't found much online to help me either. Can anyone give me a general step-by-step on how to solve these types of problems? I would really appreciate any help and direction on how to solve this. Thanks in advance!

 

[jvicens]

First of all, don't worry if you feel lost with this type of problem, it can be challenging but with some practice and understanding it can become easier.

Let's start by rewriting the given equation as:

Vx * u_x + Vy * u_y + Vz * u_z = 0

Where Vx, Vy, and Vz are the components of the vector V=(1,1,z).

Next, we can use the chain rule to rewrite u_x and u_y in terms of t, since u is a function of r(t).

u_x = u_r * r_x = u_r * 1 = u_r
u_y = u_r * r_y = u_r * 0 = 0

Similarly, we can rewrite u_z in terms of t using the chain rule:

u_z = u_r * r_z = u_r * cos(t)

Now, plugging these into our original equation, we get:

u_r + u_r * cos(t) = 0

Solving for u_r, we get:

u_r = 0

This means that u is a constant function, and we can rewrite it as:

u(x,y,z) = C

Where C is some constant.

Finally, we can use the given condition u(r(t)) = constant to solve for the value of C. Plugging in the coordinates of r(t), we get:

u(t,0,sin(t)) = C

Since u is a constant function, this means that u is equal to C for all values of t, y, and z. Therefore, the constant C is simply the value of u at any point, for example:

C = u(0,0,0) = (0, 0, sin(0)) = (0, 0, 0)

So the solution to this problem is:

u(x,y,z) = (0, 0, 0)

I hope this helps and gives you a better understanding of how to approach these types of problems. If you have any further questions, don't hesitate to ask. Good luck with your grad level course!