Differential geometry question

[Pyroadept]

Homework Statement

A function F of n real variables is called homogeneous of degree r if it satisfies
F(tx_1, tx_2, ..., tx_n) = (t^r)F(x_1,x_2,...,x_n)

By differentiation with respect to t, show that a function F is an eigenfunction of the operator:

x^1 ∂/∂x_1 + ... + x^n ∂/∂x_n

and find the eigenvalue

Homework Equations

The Attempt at a Solution

Here's what I've done so far:

Differentiating with respect to t:

dF(tx)/dt = rt^(r-1)F(x)

But what do they mean by asking me to show it is an eigenfunction of the operator, and how would I go about calculating the eigenvalue? Any hints in the right direction would be super!

Thanks!

 

[mighty2000]

Thank you for your post. To show that a function F is an eigenfunction of the given operator, we need to show that when the operator acts on F, it gives a multiple of F. In other words, we need to show that:

(x^1 ∂/∂x_1 + ... + x^n ∂/∂x_n)F(x_1, x_2, ..., x_n) = λF(x_1, x_2, ..., x_n)

where λ is the eigenvalue.

From your attempt, we have:

dF(tx)/dt = rt^(r-1)F(x)

We can rewrite this as:

dF(tx)/dx_1 = (rt^(r-1)F(x))/x_1

= t^(r-1)F(x)

= t^(r-1)(x^1 ∂F/∂x_1)

= (x^1 ∂F/∂x_1)t^(r-1)

Similarly, we can show that:

dF(tx)/dx_2 = (x^2 ∂F/∂x_2)t^(r-1)

...

dF(tx)/dx_n = (x^n ∂F/∂x_n)t^(r-1)

Therefore, we have:

(x^1 ∂/∂x_1 + ... + x^n ∂/∂x_n)F(x_1, x_2, ..., x_n) = (x^1 ∂F/∂x_1 + ... + x^n ∂F/∂x_n)t^(r-1)

= t^(r-1)(x^1 ∂F/∂x_1 + ... + x^n ∂F/∂x_n)

= t^(r-1)F(tx_1, tx_2, ..., tx_n)

= (t^(r-1))^r F(x_1, x_2, ..., x_n) (using the given condition)

= t^(r^2-r) F(x_1, x_2, ..., x_n)

Therefore, we have shown that F is an eigenfunction of the given operator with eigenvalue t^(r^2-r).

I hope this helps. Let me know if you have any further questions.