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A function is said to be homogeneous of degree $n$ if $f(\gamma x, \gamma y)=\gamma^{n} f(x, y)$ Similarly, a function of three variables is homogenous of degree $n$ if $f(\gamma x, \gamma y, \gamma z)=\gamma^{n} f(x, y, z) .$ Determine which of the following functions is homogenous, and if it is, give its degree.$$f(x, y)=\frac{x^{2}+y^{2}}{x^{3} y^{3}}$$

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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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so determine if this is going to be homogeneous of degree in, we can go ahead and first just started the left side of this equation and then see if we can force it to look like the right side. So let's go ahead and do that. So I have f of gamma X camera. Why? And then this is equal to so the gamma about why gamma X squared plus gamma y squared all over gamma Not why I m a x cute camera. Why cute? So now noticing the numerator weaken factor out a gamma squared from each Once we distribute that So it be camera squared, X squared plus y squared then all over. Well, in the denominator, we could distribute the three so we have to gamma cubes being multiplied together. Is that being gamma to the six? I don't know why I'm having such a hard time. Bring Gammas and keep writing. Six is, um bear that will be execute. Why cute And now notice this here on the right side is f of X is what we started with. And then if we were to go ahead and rewrite this here, that would be gamma to the negative worth times of X Y if we come up here and look, this is in the form that we were looking for, so this will be homogeneous, so yes, and it is a degree, uh, negative for

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