Rewriting the derivative of a homogenous function demonstration

In summary, homogeneous functions of degree r with mixed partial derivatives of all orders can be generalized by starting from the given equation and deriving both sides with respect to t, then setting t = 1. The generalization to all derivatives is straightforward.
  • #1
bobbarker
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Homework Statement


Suppose that f=f(x1,x2,...,xn) is a homogeneous function of degree r with mixed partial derivative of all orders. Show that

XZivX.png


Can this be generalized?

Homework Equations



We say that a function f is homogeneous of degree r if there exists r such that f(tx1,tx2,...,txn) = tr * f(x1,x2,...,xn)

The Attempt at a Solution


I'm pretty lost on how to show these two. I understand that this is kind of like taking the "second derivative" of f with respect to t, but how do I introduce the t into the sum terms?
 
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  • #2
bobbarker said:
We say that a function f is homogeneous of degree r if there exists r such that f(tx1,tx2,...,txn) = tr * f(x1,x2,...,xn)

Just start from this equation and derive both sides with respect to t, then set t = 1. The generalization to all derivatives is obvious to you, isn't it?
 

1. What is a homogenous function?

A homogenous function is a mathematical function where all of its variables have the same degree. This means that if you were to multiply all of the variables in the function by a constant, the function would remain unchanged.

2. Why is it important to know how to rewrite the derivative of a homogenous function?

Knowing how to rewrite the derivative of a homogenous function is important because it allows us to simplify the derivative and make it easier to work with. This can be especially useful when solving complex mathematical problems.

3. How do you rewrite the derivative of a homogenous function?

The derivative of a homogenous function can be rewritten using the Euler's Homogeneous Function Theorem. This theorem states that the derivative of a homogenous function is equal to its degree times the function itself, divided by its variables.

4. Can you provide an example of rewriting the derivative of a homogenous function?

Sure, let's say we have a homogenous function f(x,y,z) = x^2yz. To rewrite its derivative, we would use the formula: df/dt = f(x,y,z) + x(d/dx)f(x,y,z) + y(d/dy)f(x,y,z) + z(d/dz)f(x,y,z). Plugging in our function, we get: df/dt = x^2yz + x(2xyz) + y(x^2z) + z(x^2y). This can be simplified to: df/dt = 2f(x,y,z).

5. Are there any other important concepts related to rewriting the derivative of a homogenous function?

One important concept to keep in mind is that the derivative of a homogenous function is always a homogenous function of one degree less. In other words, if the original function is of degree n, its derivative will be of degree n-1. This can be useful when solving problems involving multiple derivatives of a homogenous function.

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