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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 72

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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so if you haven't done exercise 72 years, I would go do that first. Because in that, we actually figure out that, uh, this equation here is of degree six, and I'm just gonna kind of use that a supposed toe going through how we got that again. Um, but in order for us to kind of verify this, we just need to start on the left side and then kind of show. This will be equal to the right. So over here, the right side. Since this is of degree six, it would be six times what we have is actually, six times. That would end up being 18 x y squared C cute. So this is what we should end up getting when we do this. Um, so first, we'll just need to figure out what our partials Then we could go from there, so I'll do those in the corner here. So are partial. This with respect to X? Remember, we assume winds at your constants, so that would just be three wise. Where he cute on partial With respect to why? So we would use power rule if y squared assuming accent see, or Constance that would be six x y z cute and then our personal disrespect. Dizzy X y constant Z Q. They'll give us nine x y squared Z squared, so that would just come over here and plug all of those in so X times ffx would give us three x wise Where z cute. Why Times f of liar f partial with Specter Lie is going to be six um, X y squared Z cubed and NZ times. The partial with prospective Z would be nine x y squared z cute and now notice. All of these have the same a term. So we can just add the coefficients, which is going to give us 18 x y squared C cube, which is equal to this over here, so it looks like it checks out.

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