Let $g(x,y)=c$ and $f(x,y)=d$. Taking the derivative of $f$ and $g$ with respect to $t$ gives:

$\displaystyle\frac{\partial f}{\partial t}=\frac{\partial f}{\partial x}x^{{\prime}}(t)+\frac{\partial f}{\partial y}y^{{\prime}}(t)=0$

and

$\displaystyle\frac{\partial g}{\partial t}=\frac{\partial g}{\partial x}x^{{\prime}}(t)+\frac{\partial g}{\partial y}y^{{\prime}}(t)=0$

By letting $\vec{r}=x(t)\hat{i}+y(t)\hat{j},$ the partial derivatives can be rewritten as follows:

$\displaystyle\frac{\partial f}{\partial t}=\operatorname{grad}f\cdot\vec{r^{{\prime}}};$  $\displaystyle\frac{\partial g}{\partial t}=\operatorname{grad}g\cdot\vec{r^{{\prime}}}$

This implies that $\operatorname{grad}f\times\operatorname{grad}g=0,$ thus $\operatorname{grad}f=\lambda\operatorname{grad}g.$ Now this equation can be rewritten as $f_{x}\hat{i}+f_{y}\hat{j}=\lambda\left(g_{x}\hat{i}+g_{y}\hat{j}\right).$ Since $\mathbb{R}^{n}\mapsto\mathbb{R},$ this equation can be separated into two new equations:

$\displaystyle f_{x}=\lambda g_{x};\quad f_{y}=\lambda g_{y}$

Using the above equations, a new function, $F$, can be defined:

$\displaystyle F(x,y,\lambda)=f(x,y)-\lambda g(x,y)$

which can be generalized as:

$\displaystyle F(x,y,\lambda)=f(x,y)-\sum_{{i=1}}^{m}\lambda_{i}\left[g_{i}(x,y)\right].$