Convexity of a function defined in spherical coordinates by $r=f(\theta,\phi)$

Question

In the context of defining a yield surface for materials (the point at which a material changes its behavior from elastic to plastic deformation), a convex surface is to be defined in the stress domain $(\sigma_{xx},\sigma_{yy},\sigma_{xy})$, which in this problem can be considered the $(x,y,z)$ 3D space. The yield surface is defined however in spherical corrdinates using $r=f(\theta,\phi)$. So finally, we have a surface defined in 3D by its radius in spherical coordinates using $r=f(\theta,\phi)$, with $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$. I need to demonstrate if this surface is convex. I know that we need to compute the Hessian matrix, which should be a positive defined matrix. However, I can't find how to compute this Hessian anywhere for a surface defined using $r=f(\theta,\phi)$. This is not posted anywhere yet. Thank you for your help.

Answer 1

You can compute the second fundamental form of any parametrized surface in space using the method outlined here:Second Fundamental Form (It is positive definite iff the surface is convex.)

In your case the parameters are $(u,v)= (\theta, \phi)$ and the position vector in space is $\vec r (u,v)= <x,y,z> =<\rho \cos \theta \sin \phi, \rho \sin \theta \sin \phi, \rho \cos \phi>$ where $\rho=f(\phi, \theta)$ is your given function. It will be a rather tedious calculation. (Note that I used the "mathematicians' conventions" regarding the spherical coordinates $\phi$ and $\theta$. You may wish t swap their roles.)

I would split this as $\vec r= f \sin \phi < \cos \theta, \sin \theta, 0>+ f \cos \phi<0,0,1>$ and imagine traveling on an arbitrary parametrized path on this surface ( taking $\phi$ and $\theta$ to be arbitrary functions of a time parameter $t$) and then compute the velocity and acceleration $\ddot r$ using the Chain and Product rule. The surface is convex iff the component of acceleration in the direction of the outward normal vector to the surface is always positive. (Note that this method avoids the tedium of normalizing the unit normal as referenced in the Wikipedia link.)

For additional motivation for why this acceleration calculation reveals the convexity of the surface see this related post: enter link description here

P.S. To find the direction of the surface normal, note that it is the cross product of $ \vec r_u$ and $\vec r_v$.

FWIW. I ran the calculation through a symbolic computer algebra system and got this formula for the determinant of the Hessian matrix. (Its positivity is what ensures convexity). I also went ahead and swapped the letters $\theta$ and $\phi$ since that appears to be your preferred convention. Brace yourself for the mess of nested parentheses.) Condolences! Note that the superscripts indicate various partial derivatives.

$$\frac{1}{2} f(\theta ,\phi )^3 \left(-2 \sin ^2(\theta ) f(\theta ,\phi )^2 \left(f^{(0,2)}(\theta ,\phi )+\sin (\theta ) \left(\sin (\theta ) f^{(2,0)}(\theta ,\phi )+\cos (\theta ) f^{(1,0)}(\theta ,\phi )\right)\right)+f(\theta ,\phi ) \left(2 \sin (2 \theta ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi )+(1-3 \cos (2 \theta )) f^{(0,1)}(\theta ,\phi )^2+2 \sin ^2(\theta ) \left(-f^{(1,1)}(\theta ,\phi )^2+f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi )+2 \sin ^2(\theta ) f^{(1,0)}(\theta ,\phi )^2+\sin (\theta ) \cos (\theta ) f^{(2,0)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )\right)\right)-4 \sin (\theta ) \left(-2 \sin (\theta ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) f^{(0,1)}(\theta ,\phi )+f^{(0,1)}(\theta ,\phi )^2 \left(\sin (\theta ) f^{(2,0)}(\theta ,\phi )+2 \cos (\theta ) f^{(1,0)}(\theta ,\phi )\right)+\sin (\theta ) f^{(1,0)}(\theta ,\phi )^2 \left(f^{(0,2)}(\theta ,\phi )+\sin (\theta ) \cos (\theta ) f^{(1,0)}(\theta ,\phi )\right)\right)+2 \sin ^4(\theta ) f(\theta ,\phi )^3\right)$$

If you prefer it non-factored, this is that same determinant:

$$-\cos ^4(\phi ) f(\theta ,\phi )^4 f^{(0,1)}(\theta ,\phi )^2 \cos ^6(\theta )-f(\theta ,\phi )^4 \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 \cos ^6(\theta )-2 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 \cos ^6(\theta )-4 f(\theta ,\phi )^3 \sin (\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos ^5(\theta )-8 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin (\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos ^5(\theta )-4 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin (\theta ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos ^5(\theta )+2 f(\theta ,\phi )^4 \sin (\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^5(\theta )+4 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin (\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^5(\theta )+2 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin (\theta ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^5(\theta )-2 f(\theta ,\phi )^3 \sin ^2(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^4(\theta )-2 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^2(\theta ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^4(\theta )-4 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^2(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^4(\theta )-f(\theta ,\phi )^4 \sin ^2(\theta ) \sin ^4(\phi ) f^{(1,1)}(\theta ,\phi )^2 \cos ^4(\theta )-\cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^2(\theta ) f^{(1,1)}(\theta ,\phi )^2 \cos ^4(\theta )-2 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^2(\theta ) \sin ^2(\phi ) f^{(1,1)}(\theta ,\phi )^2 \cos ^4(\theta )-f(\theta ,\phi )^5 \sin ^2(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) \cos ^4(\theta )-\cos ^4(\phi ) f(\theta ,\phi )^5 \sin ^2(\theta ) f^{(0,2)}(\theta ,\phi ) \cos ^4(\theta )-2 \cos ^2(\phi ) f(\theta ,\phi )^5 \sin ^2(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) \cos ^4(\theta )+4 f(\theta ,\phi )^3 \sin ^2(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^4(\theta )+4 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^2(\theta ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^4(\theta )+8 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^2(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^4(\theta )-2 f(\theta ,\phi )^3 \sin ^2(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi ) \cos ^4(\theta )-2 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^2(\theta ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi ) \cos ^4(\theta )-4 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^2(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi ) \cos ^4(\theta )+f(\theta ,\phi )^4 \sin ^2(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^4(\theta )+\cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^2(\theta ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^4(\theta )+2 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^2(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^4(\theta )-2 f(\theta ,\phi )^3 \sin ^3(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi )^3 \cos ^3(\theta )-2 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^3(\theta ) f^{(1,0)}(\theta ,\phi )^3 \cos ^3(\theta )-4 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^3(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi )^3 \cos ^3(\theta )-f(\theta ,\phi )^5 \sin ^3(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi ) \cos ^3(\theta )-\cos ^4(\phi ) f(\theta ,\phi )^5 \sin ^3(\theta ) f^{(1,0)}(\theta ,\phi ) \cos ^3(\theta )-2 \cos ^2(\phi ) f(\theta ,\phi )^5 \sin ^3(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi ) \cos ^3(\theta )-8 f(\theta ,\phi )^3 \sin ^3(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos ^3(\theta )-8 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^3(\theta ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos ^3(\theta )-16 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^3(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos ^3(\theta )+4 f(\theta ,\phi )^4 \sin ^3(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^3(\theta )+4 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^3(\theta ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^3(\theta )+8 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^3(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^3(\theta )+f(\theta ,\phi )^4 \sin ^3(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^3(\theta )+\cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^3(\theta ) f^{(1,0)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^3(\theta )+2 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^3(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^3(\theta )+\cos ^4(\phi ) f(\theta ,\phi )^6 \sin ^4(\theta ) \cos ^2(\theta )+f(\theta ,\phi )^6 \sin ^4(\theta ) \sin ^4(\phi ) \cos ^2(\theta )+2 \cos ^2(\phi ) f(\theta ,\phi )^6 \sin ^4(\theta ) \sin ^2(\phi ) \cos ^2(\theta )+3 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) f^{(0,1)}(\theta ,\phi )^2 \cos ^2(\theta )+3 f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 \cos ^2(\theta )+6 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 \cos ^2(\theta )+2 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) f^{(1,0)}(\theta ,\phi )^2 \cos ^2(\theta )+2 f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^2(\theta )+4 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^2(\theta )-4 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^4(\theta ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^2(\theta )-4 f(\theta ,\phi )^3 \sin ^4(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^2(\theta )-8 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^4(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2 \cos ^2(\theta )-2 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) f^{(1,1)}(\theta ,\phi )^2 \cos ^2(\theta )-2 f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^4(\phi ) f^{(1,1)}(\theta ,\phi )^2 \cos ^2(\theta )-4 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^2(\phi ) f^{(1,1)}(\theta ,\phi )^2 \cos ^2(\theta )-2 \cos ^4(\phi ) f(\theta ,\phi )^5 \sin ^4(\theta ) f^{(0,2)}(\theta ,\phi ) \cos ^2(\theta )-2 f(\theta ,\phi )^5 \sin ^4(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) \cos ^2(\theta )-4 \cos ^2(\phi ) f(\theta ,\phi )^5 \sin ^4(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) \cos ^2(\theta )+8 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^4(\theta ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^2(\theta )+8 f(\theta ,\phi )^3 \sin ^4(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^2(\theta )+16 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^4(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos ^2(\theta )-\cos ^4(\phi ) f(\theta ,\phi )^5 \sin ^4(\theta ) f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )-f(\theta ,\phi )^5 \sin ^4(\theta ) \sin ^4(\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )-2 \cos ^2(\phi ) f(\theta ,\phi )^5 \sin ^4(\theta ) \sin ^2(\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )-4 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^4(\theta ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )-4 f(\theta ,\phi )^3 \sin ^4(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )-8 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^4(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )+2 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )+2 f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )+4 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^4(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos ^2(\theta )-2 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^5(\theta ) f^{(1,0)}(\theta ,\phi )^3 \cos (\theta )-2 f(\theta ,\phi )^3 \sin ^5(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi )^3 \cos (\theta )-4 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^5(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi )^3 \cos (\theta )-\cos ^4(\phi ) f(\theta ,\phi )^5 \sin ^5(\theta ) f^{(1,0)}(\theta ,\phi ) \cos (\theta )-f(\theta ,\phi )^5 \sin ^5(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi ) \cos (\theta )-2 \cos ^2(\phi ) f(\theta ,\phi )^5 \sin ^5(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi ) \cos (\theta )-4 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^5(\theta ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos (\theta )-4 f(\theta ,\phi )^3 \sin ^5(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos (\theta )-8 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^5(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(1,0)}(\theta ,\phi ) \cos (\theta )+2 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^5(\theta ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos (\theta )+2 f(\theta ,\phi )^4 \sin ^5(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos (\theta )+4 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^5(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi ) \cos (\theta )+\cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^5(\theta ) f^{(1,0)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos (\theta )+f(\theta ,\phi )^4 \sin ^5(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos (\theta )+2 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^5(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi ) \cos (\theta )+\cos ^4(\phi ) f(\theta ,\phi )^6 \sin ^6(\theta )+f(\theta ,\phi )^6 \sin ^6(\theta ) \sin ^4(\phi )+2 \cos ^2(\phi ) f(\theta ,\phi )^6 \sin ^6(\theta ) \sin ^2(\phi )+2 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) f^{(0,1)}(\theta ,\phi )^2+2 f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2+4 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2+2 \cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) f^{(1,0)}(\theta ,\phi )^2+2 f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^4(\phi ) f^{(1,0)}(\theta ,\phi )^2+4 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^2(\phi ) f^{(1,0)}(\theta ,\phi )^2-2 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^6(\theta ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2-2 f(\theta ,\phi )^3 \sin ^6(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2-4 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^6(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi )^2-\cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) f^{(1,1)}(\theta ,\phi )^2-f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^4(\phi ) f^{(1,1)}(\theta ,\phi )^2-2 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^2(\phi ) f^{(1,1)}(\theta ,\phi )^2-\cos ^4(\phi ) f(\theta ,\phi )^5 \sin ^6(\theta ) f^{(0,2)}(\theta ,\phi )-f(\theta ,\phi )^5 \sin ^6(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi )-2 \cos ^2(\phi ) f(\theta ,\phi )^5 \sin ^6(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi )+4 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^6(\theta ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi )+4 f(\theta ,\phi )^3 \sin ^6(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi )+8 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^6(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi ) f^{(1,0)}(\theta ,\phi ) f^{(1,1)}(\theta ,\phi )-\cos ^4(\phi ) f(\theta ,\phi )^5 \sin ^6(\theta ) f^{(2,0)}(\theta ,\phi )-f(\theta ,\phi )^5 \sin ^6(\theta ) \sin ^4(\phi ) f^{(2,0)}(\theta ,\phi )-2 \cos ^2(\phi ) f(\theta ,\phi )^5 \sin ^6(\theta ) \sin ^2(\phi ) f^{(2,0)}(\theta ,\phi )-2 \cos ^4(\phi ) f(\theta ,\phi )^3 \sin ^6(\theta ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi )-2 f(\theta ,\phi )^3 \sin ^6(\theta ) \sin ^4(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi )-4 \cos ^2(\phi ) f(\theta ,\phi )^3 \sin ^6(\theta ) \sin ^2(\phi ) f^{(0,1)}(\theta ,\phi )^2 f^{(2,0)}(\theta ,\phi )+\cos ^4(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi )+f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^4(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi )+2 \cos ^2(\phi ) f(\theta ,\phi )^4 \sin ^6(\theta ) \sin ^2(\phi ) f^{(0,2)}(\theta ,\phi ) f^{(2,0)}(\theta ,\phi )$$