PX275 - A5b - definition using gradient

\vec{\nabla} f + λ \vec{\nabla} g = 0

example

\begin{matrix} f (x, y) = 1 + x y \\ g (x, y) = x^{2} + y^{2} = 1 \\ \vec{\nabla} f = y \hat{i} + x \hat{j} \\ \vec{\nabla} g = 2 x \hat{i} + 2 y \hat{j} \end{matrix}

it is known that the maxima of the function with respect to the constraint lies at $x = y = \frac{1}{\sqrt{2}}$ , and $λ = - \frac{1}{2} :$

\vec{\nabla} f = \frac{1}{\sqrt{2}} (\hat{i} + \hat{j}); \vec{\nabla} g = \frac{2}{\sqrt{2}} (\hat{i} + \hat{j})

it can be seen that:

\vec{\nabla} f = \frac{1}{2} \vec{\nabla} g

this means that the vectors, $\vec{\nabla} f$ and $\vec{\nabla} g$ point in the same direction, but have different magnitudes:

\vec{\nabla} f - \frac{1}{2} \vec{\nabla} g = 0

the following condition can be identified:

\vec{\nabla} f + λ \vec{\nabla} g = 0