Question Number 224919 by fkwow344 last updated on 12/Oct/25
$${T}\left({x},{y}\right);\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R}\:,\:\:{T}\left({x},{y}\right)=\left(\frac{{x}}{{a}}\right)^{\mathrm{2}} +\left(\frac{{y}}{{b}}\right)^{\mathrm{2}} \\ $$$${K}\left({x},{y}\right)=\frac{\mathrm{4}}{{a}^{\mathrm{2}} {b}^{\mathrm{2}} \left(\mathrm{1}+\frac{\mathrm{4}{x}^{\mathrm{2}} }{{a}^{\mathrm{4}} }+\frac{\mathrm{4}{y}^{\mathrm{2}} }{{b}^{\mathrm{4}} }\right)^{\mathrm{2}} } \\ $$$$\int_{\:\mathcal{S}} \mathrm{d}\boldsymbol{\mathrm{r}}^{\mathrm{2}} {K}=? \\ $$$$\mathcal{S};\:\mathrm{sheprical}\:\mathrm{coordinate} \\ $$$$\:\: \\ $$$$\:\mathrm{and}\:\mathrm{associate}\:\mathrm{with}\:\mathrm{Euler}\:\mathrm{characteristic}\:\chi\left({T}\right) \\ $$