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

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.$$f(x, y, z)=\frac{3 x y^{2} z^{3}}{\sqrt{x^{2}+2 y^{2}+3 z^{2}}}$$

5

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Johns Hopkins University

Baylor University

University of Nottingham

Idaho State University

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

01:50

A function is said to be h…

01:15

01:39

02:01

01:14

00:53

01:17

02:48

A function $f(x, y, z)$ is…

02:57

A function $f(x, y)$ is sa…

00:50

Determine whether the func…

00:43

01:34

The function $f$ is homoge…

01:02

01:06

01:29

01:13

Determine if each function…

02:41

A function $f$ is homogene…

so for us to figure out if it's going to be home, ingenious or not, we could just start on the left side of equation and see if we can work our way towards the right. So let's go ahead and do that. So we have, uh you have Ah, Max Gamble. Why, Gamma Z come over here and just replace all those. So be three gamma x camel. Why squared? Thank Cama. Was he cute all over. So camera X squared plus two times Gamel. Why squared plus three times gamma z squared that we have the square root of all of that? Yes. Now in the numerator we can go ahead and rewrite that as gamma to the 63 x wise where C cube and then the denominator. Well, once we distribute that, each of these terms here would have a wise we're gamma square so that we can factor that out. But then we'd have square root of gamma, um square, which would just be gamma. So we have a gamma on the outside and then square root technically absolute value. But it won't really matter in this case, Um, Web X squared plus two y squared plus three c squared. Now we could go ahead and simplify that down do gamma to the fifth and then notice what we have right here is our original function of F of X of why? And so then this implies this will be of, um, a homogeneous equation. And it has degree five, since it's Gamma to the fifth power there.

View More Answers From This Book

Find Another Textbook

04:06

Consider the graph defined by $f(x)=a x^{n},$ with $a>0$ and $n \neq-1 .$…

03:16

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

02:47

Consider the area bounded by $f(x)=x^{3}+1$ and the $x$ -axis, between $x=0$…

01:44

Determine the area of the indicated region.Region bounded by $f(x)=3 x$ …

04:25

Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{…

01:51

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

02:23

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

03:24

04:02

The partial differential equation $c^{2} u_{x x}-u_{t t}=0$ where $c$ is a c…

01:09

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