Curl Eigenvector D583Cb

1. **State the problem:** We want to find a non-zero vector function $u$ satisfying the equation $$\nabla \times u - f u = 0,$$ where $f$ and $g$ are given functions, $g$ is not constant, and $$\Delta g = 0.$$ 2. **Given information and conditions:** - $f$ and $g$ are given scalar functions. - $g$ is harmonic since $$\Delta g = 0.$$ - The equation to solve is $$\nabla \times u = f u.$$ 3. **Key observations:** - The equation $$\nabla \times u = f u$$ is a type of eigenvalue problem for the curl operator with eigenvalue $f$. - Since $g$ is harmonic, it satisfies $$\Delta g = 0.$$ 4. **Approach to find $u$: ** - From the original equation $$\nabla \times w - f w = \nabla g,$$ we see that $$\nabla \times w = f w + \nabla g.$$ - If we want to find $u$ such that $$\nabla \times u = f u,$$ one natural idea is to consider $$u = w + \nabla h,$$ where $h$ is a scalar function to be determined. 5. **Substitute $u = w + \nabla h$: ** - Compute $$\nabla \times u = \nabla \times w + \nabla \times \nabla h = \nabla \times w + 0 = f w + \nabla g.$$ - Then $$\nabla \times u - f u = (f w + \nabla g) - f (w + \nabla h) = \nabla g - f \nabla h.$$ 6. **Set the equation to zero:** - We want $$\nabla \times u - f u = 0,$$ so $$\nabla g - f \nabla h = 0 \implies \nabla g = f \nabla h.$$ 7. **Solve for $h$: ** - If $f$ is nonzero, then $$\nabla h = \frac{1}{f} \nabla g.$$ - Integrate to find $h$: $$h = \int \frac{1}{f} \nabla g \cdot d\mathbf{x} + C,$$ where $C$ is a constant. 8. **Summary:** - Given $w$ satisfying $$\nabla \times w - f w = \nabla g,$$ and $g$ harmonic, - Define $$u = w + \nabla h,$$ where $$\nabla h = \frac{1}{f} \nabla g.$$ - Then $u$ satisfies $$\nabla \times u = f u,$$ a non-zero solution if $w$ and $\nabla h$ are non-zero. **Final answer:** $$\boxed{u = w + \nabla h, \quad \text{where} \quad \nabla h = \frac{1}{f} \nabla g.}$$