# 'Definition' of the Lagrange multipliers

I do not fully understand the 'definition' of the Lagrange multipliers. I do understand that a maximum occurs when the constraint and the objective function are tangent to eachother. However, I do not understand why this implies that the $\nabla{f}=\lambda\times\nabla{g}$. Why is it not true that the gradient of $f$ IS EQUAL to the gradient of $g$? Doesn't the fact that the level curves are parallel imply that the derivatives are equal and thus that the gradients are equal (and not a multiple of eachother)?

• For example, optimizing under the constraint $x^2+y^2 = 1$ is the same as optimizing under the constraint $2x^2+2y^2 = 2$, but $x^2+y^2$ and $2x^2+2y^2$ have different gradients. The direction of the level curves determine the direction of the gradient, but not the magnitude. (It is the spacing that determines magnitude, as Nick answered.) – Jonas Meyer Nov 5 '12 at 17:03
• Thank you for the clarification :). – dreamer Nov 5 '12 at 18:19