Euler's Theorem of Homogeneous Function

Euler's Theorem of Homogeneous Functions

Euler's Theorem of Homogeneous Functions

Euler's theory of equivalent functions clarifies a key concept in statistical analysis, showing the relationship between equivalent functions. Discover deeper insights into the scaling behavior of these functions and their impact on mathematical models, optimization, and applications. Unpack the basic principles of Euler's theorem and dive clearly and deeply into the fascinating world of identical functions.

If \(z\) be a homogeneous function of \(x, y\) of order \(n\), then \(x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=nz\) all

Proof: Since \(z\) is a homogeneous function of \(x, y\) of order \(n\). \(z=x^{n}f(\frac{y}{x}) \dots (1)\)

\(\frac{\partial z}{\partial x}=n~x^{n-1}f(\frac{y}{x})+x^{n}.f^{\prime}(\frac{y}{x}).(-\frac{y}{x^{2}}).\) or \(\frac{\partial z}{\partial x}=n~x^{n-1}f(\frac{y}{x})-y~x^{n-2}f^{\prime}(\frac{y}{x})\)

Multiplying both sides by \(x\), we get, \(x\frac{dz}{dx}=nx^{n}f(\frac{y}{x})-yx^{n-1}f^{\prime}(\frac{y}{x}) \dots (2)\)

Again from (1), we have, \(\frac{\partial z}{\partial y}=x^{n}f^{\prime}(\frac{y}{x}),\frac{1}{x}\) or \(\frac{\partial z}{\partial x}=x^{n-1}f^{\prime}(\frac{\nu}{x}) \dots (3)\)

Multiplying both sides by \(y\), we get, \(y\frac{\partial z}{\partial x}=y~x^{n-1}f^{\prime}(\frac{y}{x})\).

Adding (2) and (3), we get, \(x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=nz\)

Comments

Post a Comment

Popular posts from this blog

Fundamental Theorem of Algebra in Complex Analysis

Image
  In this page, we are going to analysis the important topic or theorem in Complex Analysis.    Fundamental Theorem of Algebra in Complex Analysis is a very important topic to understand that can enhance your Higher Mathematics knowledge. Fundamental Theorem of Algebra in Complex Analysis Statement: If f(z) =  a 0 z n + a 1 z n-1 + ... + a n , a 0 ≠ 0 be a polynomial in z, then f(z) = 0 has at least one root in z-plane. Proof: If possible, let f(z) = 0 has no root in z-plane. Therefore, f(z)  ≠ 0 for any value of z in z-plane. We define, because, f(z) is analytic and f(z)  ≠ 0 therefore, F(z) is analytic for all values of z. Also, |F(z)| --> 0 as |z| -->  ∞  therefore, F(z) is bounded therefore, By Liouville's Theorem, F(z) is constant. ⇒ f(z) is constant, which is contradiction [because, f(z) is not constant polynomial] therefore, f(z) = 0 has at least one root in z-plane. Explanation in YouTube:
Read more

Intersection of Two Subspaces is a Subspace but no need of Union

Image
Intersection and Union of Subspaces Intersection of Two Subspaces is a Subspace Proof: Let \( S \) and \( T \) be two subspaces of a vector space \( V \) over a field \( F \). We shall prove that \( S \cap T \) is also a subspace of \( V \). Let \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Then \( x, y \in S \) and \( x, y \in T \). Since \( S \) and \( T \) are subspaces of \( V \), we have: \[ \alpha x + \beta y \in S \] \[ \alpha x + \beta y \in T \] Therefore, \[ \alpha x + \beta y \in S \cap T \] Thus, \( \alpha x + \beta y \in S \cap T \) for all \( \alpha, \beta \in F \) and \( x, y \in S \cap T \). Hence, \( S \cap T \) is a subspace of \( V \). Union of Two Subspaces Need Not Be a Subspace Now, by giving an example we shall prove that the union of two subspaces need not be a subspace. Example: Let \( V = \mathbb{R}^3 = \{(x, y, z) : x, y, z \in \mathbb{R}\} \) be a vector space over the field \( \mathbb{R} \) of all rea...
Read more

Triangle Inequality in Complex Analysis

Image
Triangle Inequality in Complex Analysis What is the Triangle Inequality? In Complex Analysis, it states that any two complex number z 1 and z 2 , the equality holds: |z 1 + z 2 | ≤ |z 1 | + |z 2 | Here, |z| denotes the modulus of the complex number, and the modulus of complex number z = x + y i , where x is the real number and y i is the imaginary number. So, in complex analysis is defined as, |z| = √(x 2 + y 2 ) What is the Triangle Inequality number theory? In Number Theory, the Triangle Inequality is discussed in the context of the absolute value function over the integers or the real number. It states that for any real number a and b, the Inequality holds: |a + b| ≤ |a| + |b| Proof of Triangle Inequality in Number Theory. Wha...
Read more