I have the following set of equations:
1:
$f(x,y,z) = f_xx+f_yy+f_zz + d_1$
$g(x,y,z) = g_xx+g_yy+g_zz + d_2$
where $ f_x,f_y,f_z, d_1, g_x,g_y,g_z, d_2$ are known scalar constants, and
$x,y,z \subset(-1e5,1e5)$
2:
$r(f,g) = \sqrt{f^2+g^2}$
$N_c(f,g) = \sqrt{(ff_x+gg_x)^2 + (ff_y+gg_y)^2 + (ff_z+gg_z)^2}$
$N_s = \sqrt{(f_yg_z-f_zg_y)^2+(f_zg_x-f_xg_z)^2+(f_xg_y-f_yg_x)^2}$
3:
$v_{zd} = \frac{-V(ff_z+gg_z)tanh(kr)}{N_c} + \frac{V(f_xg_y-f_yg_x)sech(kr)}{N_s}$
$v_{xd} = \frac{-V(ff_x+gg_x)tanh(kr)}{N_c} + \frac{V(f_yg_z-f_zg_y)sech(kr)}{N_s}$
where $k$ is unknown number which I want to find.
4:
$\chi_d = atan2(v_{zd}, v_{xd})$
$B_x = \frac{\partial \chi_d}{\partial x}$
$B_y = \frac{\partial \chi_d}{\partial y}$
$B_z = \frac{\partial \chi_d}{\partial z}$
I've been informed to find the value of $k$ from the last following two inequalities:
5:
$\sqrt{B_x^2 + B_z^2} \le \frac{7(1-\alpha)\dot{\chi}_{max}}{10V}$
$\lvert{B_y}\rvert \le \frac{\alpha\dot{\chi}_{max}}{V}$
with $0 < \alpha < 1$
and $\dot{\chi}_{max}, V$ are known constants.
Just to make things little more easier, let $\alpha = 0.5, \dot{\chi}_{max} = 0.3, V = 50$
The ultimate problem here is that $B_x, B_y, B_z$ do not have explicit symbolic equations because I have to compute them numerically since their direct symbolic equations are too complicated even for the Matlab symbolic engine.
So the question is: how to numerically solve equations (5:) for the variable $k$ in Matlab/Ocatve?