Partial derivatives Q involving homogeneity of degree n

In summary, to show that if f is homogeneous of degree n, then x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y), use the Chain Rule to differentiate f(tx,ty) with respect to t. Then, set a = tx and b = ty and use the derivative of t^nf(x,y) to find the derivative of the right hand side of (1). This will lead to the desired result.
  • #1
kostoglotov
234
6

Homework Statement



Show that if f is homogeneous of degree n, then

[tex] x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y) [/tex]

Hint: use the Chain Rule to diff. f(tx,ty) wrt t.

2. The attempt at a solution

I know that if f is homogeneous of degree n then [tex] t^nf(x,y) = f(tx,ty) [/tex]

But I'm really at a conceptual loss here.

I've tried the hint, I let f(tx, ty) = f(a,b) so a = tx and b = ty then

[tex] \frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}\frac{\partial a}{\partial t}+\frac{\partial f}{\partial b}\frac{\partial b}{\partial t} \\
\frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}x + \frac{\partial f}{\partial b}y [/tex]

And I have tried looking at

[tex] \frac{d}{dt}t^nf(x,y) = nt^{n-1}f(x,y) [/tex]

But beyond this I just cannot see what I need to do.

Thanks.
 
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  • #3
That's a good start. I'll call the first equation in your attempt (1) and the last one (2).
Now in (2) you are taking the derivative (with respect to t) of the left hand of (1). Do the same thing with the right hand and see where that leads you.
 

Related to Partial derivatives Q involving homogeneity of degree n

What is a partial derivative?

A partial derivative is a mathematical concept used in multivariable calculus to describe the rate of change of a function with respect to one of its input variables, while keeping all other input variables constant.

What is homogeneity of degree n?

Homogeneity of degree n refers to a property of a function where all of its input variables are raised to the same power n. This means that if all the input variables are multiplied by a constant, the output of the function will also be multiplied by that same constant raised to the power n.

What is the relationship between partial derivatives and homogeneity of degree n?

If a function is homogeneous of degree n, then its partial derivatives of all orders are also homogeneous of degree n-1. This means that the degree of homogeneity decreases by 1 with each partial derivative taken.

Why is homogeneity of degree n important in partial derivatives?

Homogeneity of degree n is important in partial derivatives because it allows us to simplify the calculations and make use of certain properties of homogeneous functions. It also helps in solving optimization problems where the goal is to find the maximum or minimum of a function.

What are some real-world applications of partial derivatives involving homogeneity of degree n?

Partial derivatives involving homogeneity of degree n have many applications in economics, physics, and engineering. For example, in economics, they are used to analyze production functions and marginal rates of substitution. In physics, they are used in thermodynamics to describe the behavior of homogeneous systems. In engineering, they are used in optimization problems and to model physical systems with multiple input variables.

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