Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

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 68

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Johns Hopkins University

Oregon State University

University of Michigan - Ann Arbor

Boston College

Lectures

12:15

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.

04:14

01:13

Euler's theorem state…

02:07

01:57

A function is said to be h…

01:15

01:39

01:50

02:01

so if you haven't done number 68 yet, I would go to that. Because in that question, we end up finding out that this is of degree one half on. I won't go through how we got that again. Um, so you just go watch that video if you haven't yet. But now for us to verify the serum, um, we just need to find the partial respect, X partial perspective. Why? And then plug these in over here and then see if this is equal to Gamma of ffx. So in this case, this would be one half times route X plus. Why? This is what we're trying to see. If it is equal to honor eso, let's go ahead and find our partials. I'll do that over here on the side or partial with respect to X. Remember, this is actually do the one half power, so we would dio change rules would be one half x plus y to the negative one half times the derivative inside, which is just gonna be one s O. That would be one over to root X plus why and now over here, if we were to do the same thing for the partial with respect to why we should actually get the same thing. So it will follow the same steps and we end up with two over Route X plus Why, Okay, now, um, spill the question mark here. Let's go ahead and plug everything on the left side and see if we get this one half route X plus why? And so we would have so x times one over to root X plus y plus why times one over to root X plus y. And now we can go ahead and combine these in the numerator so it be X plus y over to root X plus y now X plus y over Route X plus why that would just be root X plus y in the numerator on. Then I'll pull that one half out. Fronts would be one half route X plus y, which is equal to gamma f of X y. So. But there, um, checks out

View More Answers From This Book

Find Another Textbook

03:45

Determine the area of the indicated region.Region bounded between $f(x)=…

04:10

Evaluate the double integral $\int_{R} \int f(x, y) d A$ over the indicated …

02:59

Consider the parabola $f(x)=a x^{2},$ with $a>0 .$ When $x=b,$ call the $…

04:56

Evaluate $\int_{R} \int f(x, y) d A$ for $R$ and $f$ as given.(a) $f(x, …

Euler's theorem states that for smooth homogenous functions of degree $…

01:51

Find the volume of the solid bounded above by the surface $z=f(x, y)$ and be…

A function is said to be homogeneous of degree $n$ if $f(\gamma x, \gamma y)…

05:10

$u(x, y)$ is said to be harmonic if it is a solution to the partial differen…

02:16

Consider the area bounded by $f(x)=x^{2}+1$ and the $x$ -axis, between $x=0$…

01:02

Determine the region $R$ determined by the given double integral.$$\int_…