- #1
Vicfred
- 14
- 0
Homework Statement
I want to calculate [tex]\nabla\times[\vec{F}(r)][/tex] and [tex]\nabla^2[\vec{F}(r)][/tex] where F if a function that depends of r, and [tex]r = \sqrt{x^2+y^2+z^2}[/tex]
Homework Equations
[tex]1)\nabla \times \vec A = \left|\begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ \\ A_x & A_y & A_z \end{matrix}\right|[/tex]
[tex]2)\nabla^2\vec A = \nabla(\nabla\cdot\vec A)-\nabla\times(\nabla\times\vec A)[/tex]
The Attempt at a Solution
[tex]\left( \frac{\partial}{\partial y}\vec F_{z}(r)-\frac{\partial}{\partial z}\vec F_{y}(r)\right)\hat{i} + \left( \frac{\partial}{\partial x}\vec F_{z}(r)-\frac{\partial}{\partial z}\vec F_{x}(r)\right)\hat{j}
+ \left( \frac{\partial}{\partial x}\vec F_{y}(r)-\frac{\partial}{\partial y}\vec F_{x}(r)\right)\hat{k}[/tex]
I need help calculating the partials I tried this:
[tex]\frac{1}{r}\left(y - z \right) + \frac{1}{r}\left(x - z \right) + \frac{1}{r}\left(x - y \right) \Rightarrow \frac{2x - 2z}{r}[/tex] What happened to the i, j, k? I don't know =P! I suppose I'm wrong for vanishing them...
When I'm trying to calculate the vectorial laplacian I get stuck when I have to calculate [tex]\nabla\times\vec {F}(r)[/tex] so...