After 1 month work on this problem, I found out following remarks:
1. If we tend to use "Brioschi formula" to calculate Gaussian curvature of a surface, we should embed it into a 2D space. For example, in the given metric, we should take r=constant to achieve a surface in theta-phi 2D space...
Rewrite the power as:
-x^{2}+2x=-(x-1)^2+1
It we define x-1=t,
-x^{2}+2x=-t^2+1
so
e^{-x^{2}+2x}=e^{-t^2+1}=e^{-t^2}e^{1}
and we also know that dx=dt, change the variable of integral and enjoy!
Well,
f(x)=x(x^{3}+2x)^{\frac{1}{2}}
If we apply the product rule, we should write:
\frac{df(x)}{dx}=(\frac{dx}{dx})[(x^{3}+2x)^{\frac{1}{2}}]+x(\frac{d[(x^{3}+2x)^{\frac{1}{2}}]}{dx})
So if we rearrange the equations...
[FONT="Georgia"][FONT="Tahoma"]Homework Statement
Assume that we have a metric like:
ds^{2}=f dr^{2}+ g d\theta^{2}+ h d\varphi^{2}
where r,\theta , \varphi are spherical coordinates.
f,g and h are some functions of r and theta but not phi.
Homework Equations
How can I calculate...