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For a scalar function with continuous third derivatives near $a$, apply the one-dimensional Taylor theorem to $g(t)=f(a+th)$. The chain rule gives $g^{\prime }(0)=Df(a)h$ and $g^{\prime \prime }(0)=h^T{H}_f(a)h$, proving the displayed formula. Here $H_f$ is the Hessian matrix. The finite polynomial part is the Multivariate Taylor polynomial; the remainder controls its accuracy.

A linear estimator in nonparametric regression at $x_0$ has the form ${\hat{m}}_n(x_0)={\sum }_i{W}_i(x_0)Y_i$, where the weights may depend on the design, $x_0$, $p$, $h$, and $K$, but not on the responses. Putand define the local polynomial Gram matrixWhen ${\lambda }_0>0$, the weighted least squares normal equations have the unique solutionTherefore ${\hat{m}}_n(x_0)={\sum }_i{W}_i{Y}_i$, where the effective kernel weight is

The identityshows that these weights exactly reproduce at $x_0$ every multivariate polynomial of total degree at most $p$. This is the required polynomial reproduction property of local polynomial regression.

Only grid points with $\Vert x_i-x_0{\Vert }_{\infty }\le h$ have nonzero weight. There are at most $(2n_{1}h+1{)}^d\le (4n_{1}h{)}^d=4^{d}n{h}^d$ such points because $n_{1}h\ge 1/2$. On this support,so the operator norm bound $\Vert B_n^{-1}{\Vert }_{\mathrm{op}}={\lambda }_0^{-1}$ givesBecause the errors are independent random variables with the stated variance bounds,Thus ${\gamma }_1=d$.

Let $T_{x_0}$ be the Multivariate Taylor polynomial of $m$ at $x_0$ through total degree ${\beta }_0$. The assumed Hölder continuity of the derivatives andgive the Taylor remainder boundPolynomial reproduction cancels $T_{x_0}$ in the bias. The same support count and weight bound giveConsequentlyso ${\gamma }_2=\beta$.

The partial derivatives are $f_x=e^{xy}(1+xy)$ and $f_y=x^2{e}^{xy}$. At $(1,1)$ they have values $2e$ and $e$. The first-order Multivariate Taylor polynomial therefore gives the changeThus the requested linear approximation is $e(2\delta x+\delta y)$; it is the change from $f(1,1)=e$, not the new function value itself.