the matrix $ \begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix} $ = $ \begin{bmatrix} 1&0 \\ 0&r{(\gamma )^2}\end{bmatrix}$
what i did is $$\nabla^2f=\frac{1}{r(\gamma)} [\partial _\gamma((r(\gamma ) \partial_{\gamma})+\partial_{\theta}(r(\gamma)^3\partial_{\theta})])$$
$$=\frac{1}{r(\gamma)}[\dot{r}{(\gamma)\frac {\partial}{\partial \gamma} +r(\gamma) \frac{\partial^2}{\partial \gamma^2}+r(\gamma)^3 \frac {\partial ^2}{\partial \theta^2}}] $$ but in the book it is $$ =\frac{1}{r(\gamma)}[\dot{r}{(\gamma)\frac {\partial}{\partial \gamma} +r(\gamma) \frac{\partial^2}{\partial \gamma^2}+1/r(\gamma)\frac {\partial ^2}{\partial \theta^2}}]$$. please tell where i am wrong
$g^{ij}$ represents the $ij$th element of $g^{-1}$ by convention, so you should have
$$\sqrt{|g|}g^{22} = r(\gamma)\cdot \frac{1}{r(\gamma)^2} = \frac{1}{r(\gamma)}$$
instead of
$$\sqrt{|g|}g^{22} = r(\gamma)\cdot r(\gamma)^2 = r(\gamma)^3$$
which is not correct. This is a fairly common mistake and one I've made a few times myself in this context. Note that the $11$ entry was unaltered (and thus you got the right expression for that part) since $g_{11} = 1$ so that $g^{11} = 1$.
In the Wiki article for the Laplace-Beltrami operator, they also mention that $g^{ij}$ represents the $ij$th entry of $g^{-1}$ in the introduction.