Homogeneous function of degree n

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Discussion Overview

The discussion revolves around the proof of properties related to homogeneous functions of degree n, particularly focusing on the differentiation of such functions and the implications of Euler's identity. Participants express confusion about specific steps in the proof and its broader connections to mathematical physics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the disappearance of the t^(n-1) term in the proof of the homogeneous function property.
  • Another participant emphasizes the importance of the proof in understanding transcendental functions and connects it to Euler's identity, suggesting a relationship between mass, velocity, and exponential functions.
  • A later post reiterates the initial confusion about the proof and attempts to clarify that the result holds for all t, specifically noting the case when t = 1.
  • There is a call for further elaboration on the proof and a challenge to demonstrate mastery in mathematical physics.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are expressions of confusion and requests for clarification, alongside assertions of the proof's significance and its connections to broader concepts.

Contextual Notes

The discussion includes unresolved questions about the mathematical steps involved in the proof and the implications of the terms used, particularly regarding the behavior of the t^(n-1) term.

Who May Find This Useful

This discussion may be of interest to those studying homogeneous functions, differential calculus, and their applications in physics and engineering, particularly in the context of transcendental functions and mathematical proofs.

clairaut
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I can't see the proof clearly.

Gradient [f(tx,ty)] dot
<d/dt (tx) , d/dt (ty)> =
d/dt [f(tx,ty)] = n * t^(n-1) * [f(x,y)]

I don't see how the t^(n-1) term disappears.

HELP
 
Physics news on Phys.org
Help me!
 
Echo!
 
This proof is fundamental to our understanding of transcendental functions.

This proof is intimately connected to Euler's identity in that mass raised to the power of a rational number should equal e raised to the velocity function as fxn of mass.

Propulsion can also be designed in a cyclic process that is as stable as the circular motion that contains the constant centripetal acceleration.

Also, the circular wheel contains innate structural stability due to the forces pushing inward toward the center that causes tangential velocity.

Euler's number is so integral to the formation of a circle that there is an operator that brings out the power placed on the base as a multiple of the original function. This operator is known as the natural logarithm.

This proof must be understood
well!

Can somebody continue on with this incredible proof?

Prove inside this physics forum who is the real master of mathematical physics.
 
clairaut said:
I can't see the proof clearly.

Gradient [f(tx,ty)] dot
<d/dt (tx) , d/dt (ty)> =
d/dt [f(tx,ty)] = n * t^(n-1) * [f(x,y)]

I don't see how the t^(n-1) term disappears.

HELP

The result is true for all [itex]t \in \mathbb{R}[/itex], so in particular it is true for [itex]t = 1[/itex]. Hence [tex] \mathbf{x} \cdot \nabla f = n 1^{n-1} f(\mathbf{x}) = n f(\mathbf{x})[/tex] as required.
 

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