State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . Verify this theorem for the functions f and g, defined by

A necessary condition for the functional dependence of two differentiable functions \( f \) and \( g \) on an open subset \( D \) of \( \mathbb{R}^n \) is that the gradients of \( f \) and \( g \) must be proportional, i.e., there exists a scalar function \( \lambda \) on \( D \) such that:

\[
______ _______ _______ ________ ________ __________.
________ _____ _______ ________ ____ __________ ___ ___.
____ _________ _____ ___ ____ __________.
_____ _______ ______ _________ ____ _____ _______ ___ ______ ______.
___ ______ ________ ____ ______ ___ ______ ________.
_____ ____ ___ _____ ____ _____.
_______ _______ ________ ______ ___ ______.
__________ _____ _____ ___ ___ _________ ______ ____ ___ ______.
______ _______ _______ _____ _________.
________ _____ _________ __________ ___ __________ ______.
____ ____ ___ __________ __________ ____.
_____ __________ ______ ______ _______ __________.
________ _________ ______ ________ ________ __________ _______.
_________ _________ _________ ___ ____ _________ ____ ___ ___ ___.
______ ____ _______ ______ __________ ________ ____ _______ ___ ________.
_______ ________ _____ ________ ________ ________ _____ ___.
__________ ____ _________ _________ ___ _____ _____.
________ _______ ________ ________ ________.
__________ _________ _____ ______ ___ _____ ___ ______ ______ ________ __________ _____.
__________ _______ __________ ________ _____ ____ ____ _______ ________ _______ ___ ____.
_____ _________ ______ ______ _______.
________ ______ ______ _______ _____ ____ _________.
____ ______ ________ _______ ____.
__________ ____ ______ ______ ____ __________ _________ ___.
_______ ____ ___ _________ ____ _________ __________ ___ _________ _________ _______ _____.
_________ _____ ___ _____ ______ _______ __________ ________ ___ ___ ______.
_____ ____ ________ ________ ____ _____ ___ ____ ______.
___ _____ _____ _____ ________ _________ ____ _____ _____ _____ __________ ______.
___ ____ _____ ____ ________ ___.
____ ____ _____ _________ ____ _____ ___ _____ ___.
_______ __________ ____ ____ ___.
_____ _____.
Get Full Answer on WhatsApp