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Euler's theorem states that for smooth homogenous functions of degree $\gamma$ that for a function of two variables,$$x f_{x}(x, y)+y f_{y}(x, y)=\gamma f(x, y)$$ and for functions of three variables, $$x f_{x}(x, y, z)+y f_{y}(x, y, z)+z f_{z}(x, y, z)=\gamma f(x, y, z)$$ if the function is homogeneous, verify it satisfies Euler's theorem.Exercise 67

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Euler's theorem state…

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A function is said to be h…

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so if you haven't done exercise 67 yet, I would go to that because in exercise 67 we actually have already found, um that the degree of this function here is just going to be one for it to be home genius. So what we need to now due to verify this is to just find a partial perspective x y and then plug everything else. So over here on the side, all kind of right thes out. So the partial with respect to X well derivative of X is going to be one derivative y zero. So the partial, that's just one partial perspective. Why? So now it's just kind of flip. So x zero, why is one then we come down here and plug visit. So be X times one plus why times one and then over here, I'll put like a little question mark So gamma. In this case, they say it's supposed to be our degree, so we do one times ffx, which would be x plus why and then over here we have just x plus why, which is indeed equal to X plus y. So this checks out

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