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As I understood, to find critical solutions for some system of equations, we assume if giving a function f() restricted to some level set of other functions gi() with the same domain and range, then its gradient vector will be perpendicular to the tangent vector of the level set at a critical points of f() and can be expressed as a linear combination of gradient vectors of functions gi() at that point. Also to get particular results, we should have enough information for our linear equation.

The book suggests to just put our linear combinations to restriction functions and solve for lambdas. That is the part that I'm struggling. I can't describe the meaning of this step and why we can acquire the answer without even using information from the distance function.

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bio grisha
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1 Answers1

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I can't describe the meaning of this step and why we can acquire the answer without even using information from the distance function.

It is using information about $f$. we have $\nabla f = (2v, 2w, 2x, 2y, 2z)$. Then the book "absorbs the $2$ into the $\lambda$'s" and just writes $(v, w, x, y, z)$ instead.

The general formula they're using here is: $$ \nabla f = \lambda_1 \nabla g_1 + \lambda_2 \nabla g_2 .$$

aras
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  • Thanks a lot, I got confused at the moment with the twos and thought that it is about other restrictions, now it is clear. – bio grisha Nov 20 '22 at 02:34