Finding Beltrami field in Cartesian coordinates

[sgregory8]

Homework Statement

Working in Cartesian coordinates (x,y,z) and given that the function g is independent of x, find the functions f and g such that: v=coszi+f(x,y,z)j+g(y,z)k is a Beltrami field.

Homework Equations

From wolfram alpha a Beltrami field is defined as v x (curl v)=0

The Attempt at a Solution

So jumping in, I compute the curl of v and the cross product of the curl of v and v:

But then the above nasty equation results and surely this would give rise to some equally nasty PDEs?

 

[haruspex]

So, that last equation gives three equations, since each component must be zero. Look at the first one, the x component. Only one term (f.f_x) is a function of x. So what do you get if you differentiate wrt x?

 

[sgregory8]

So, that last equation gives three equations, since each component must be zero. Look at the first one, the x component. Only one term (f.f_x) is a function of x. So what do you get if you differentiate wrt x?

Thanks for the tip haruspex but I'm a little confused as to what you mean. Differentiating the first component with respect to x will remove the function g(y,z) but still leave a nasty PDE.

I've found another piece of information which I think allows me to solve the problem and that is that a Beltrami field can also be denoted by curl v = kv where k>0 is just a constant. The second component then simplifies nicely where I've deduced k=1.

If I'm right I think f(x,y,z)=+-sin(z) and g(y,z)=0. This satisfies the equation. I think I initially doubted such answers as I believed that the functions f(x,y,z) and g(y,z) should contain at least y and z variables!

 

[haruspex]

Differentiating the first component with respect to x will remove the function g(y,z) but still leave a nasty PDE.

I didn't think it was nasty. Using subscript notation, I got ${f_x}^2+f_{xx}=0$. Dividing through by f_x, this integrates to produce $f^2=(Ax+B)\hat f(y,z)$.

I've found another piece of information which I think allows me to solve the problem and that is that a Beltrami field can also be denoted by curl v = kv where k>0 is just a constant. The second component then simplifies nicely where I've deduced k=1.

If I'm right I think f(x,y,z)=+-sin(z) and g(y,z)=0. This satisfies the equation. I think I initially doubted such answers as I believed that the functions f(x,y,z) and g(y,z) should contain at least y and z variables!

That sounds a very useful fact. Continuing my first principles approach, I also found g=0, but my f is a little more general: $\sqrt{C-\cos^2z}$. Maybe I missed some way of showing C=1.