Consider a continuous and differentiable function $G$ of two variables $x$ and $y$. If $\delta x$, $\delta y$ are small changes in $x$ and $y$, respectively, and $\delta G$ is the resulting small change in $G$, then by using the Taylor Series, we have \begin{equation} \delta G = \frac{\partial G}{\partial x}\delta x + \frac{\partial G}{\partial y}\delta y + \frac{1}{2} \frac{\partial^2 G}{\partial x^2}\delta x^2 + \frac{\partial^2 G}{\partial x \partial y}\delta x\delta y + \frac{1}{2} \frac{\partial^2 G}{\partial y^2}\delta y^2 + \cdots \end{equation} As the limits $\delta x$ and $\delta y$ tend to zero, why the previous equation becomes \begin{equation} dG=\frac{\partial G}{\partial x}dx + \frac{\partial G}{\partial y}dy, \end{equation} rather than \begin{equation} dG=\frac{\partial G}{\partial x}dx + \frac{\partial G}{\partial y}dy + \frac{1}{2} \frac{\partial^2 G}{\partial x^2}dx^2 + \frac{\partial^2 G}{\partial x \partial y}dxdy + \frac{1}{2} \frac{\partial^2 G}{dy^2}\delta y^2 + \cdots \end{equation}?
Thanks.
By definition, the differential $dG$ is a linear approximation to the change in $G$ near a point. Informally, this lets us ignore higher-order terms in the actual difference. More formally, $dG_{\mathbf x}$, when it exists, is the linear map such that $\delta G_{\mathbf x}[\mathbf h]=dG_{\mathbf x}[\mathbf h]+o(\|\mathbf h\|)$. Comparing this to the Taylor Series, we have $$dG(\delta x,\delta y)={\partial G\over\partial x}\delta x+{\partial G\over\partial y}\delta y,$$ while the second-degree and higher terms in $\delta x$ and $\delta y$ are subsumed into $o(\|(\delta x,\delta y)\|)$.