Math Problem Statement

  • Use Euler theorem to find 𝒙 𝒇𝒙 + 𝒚 𝒇𝒚; if a) 𝑓(𝑥, 𝑦) = √𝑥 2 + 𝑦 2

Solution

Euler's theorem states that for a homogeneous function f(x,y)f(x, y) of degree nn, the following relation holds:

xfx+yfy=nf(x,y)x f_x + y f_y = n f(x, y)

where fxf_x and fyf_y represent the partial derivatives of f(x,y)f(x, y) with respect to xx and yy, respectively. To apply this theorem, we first need to determine whether the given function f(x,y)=x2+y2f(x, y) = \sqrt{x^2 + y^2} is homogeneous, and if so, find its degree.

Step 1: Check if f(x,y)=x2+y2f(x, y) = \sqrt{x^2 + y^2} is homogeneous

A function is homogeneous of degree nn if:

f(λx,λy)=λnf(x,y)f(\lambda x, \lambda y) = \lambda^n f(x, y)

For f(x,y)=x2+y2f(x, y) = \sqrt{x^2 + y^2}, let's check the behavior when we scale xx and yy by λ\lambda:

f(λx,λy)=(λx)2+(λy)2=λ2(x2+y2)=λx2+y2f(\lambda x, \lambda y) = \sqrt{(\lambda x)^2 + (\lambda y)^2} = \sqrt{\lambda^2(x^2 + y^2)} = \lambda \sqrt{x^2 + y^2}

Thus, f(λx,λy)=λf(x,y)f(\lambda x, \lambda y) = \lambda f(x, y), meaning that f(x,y)f(x, y) is a homogeneous function of degree 11 (i.e., n=1n = 1).

Step 2: Apply Euler's theorem

Since f(x,y)f(x, y) is homogeneous of degree 1, Euler's theorem tells us that:

xfx+yfy=1f(x,y)=f(x,y)x f_x + y f_y = 1 \cdot f(x, y) = f(x, y)

Step 3: Verify by finding fxf_x and fyf_y

To confirm, we will compute the partial derivatives of f(x,y)=x2+y2f(x, y) = \sqrt{x^2 + y^2}.

  • fx=x(x2+y2)=xx2+y2f_x = \frac{\partial}{\partial x} \left( \sqrt{x^2 + y^2} \right) = \frac{x}{\sqrt{x^2 + y^2}}
  • fy=y(x2+y2)=yx2+y2f_y = \frac{\partial}{\partial y} \left( \sqrt{x^2 + y^2} \right) = \frac{y}{\sqrt{x^2 + y^2}}

Now compute xfx+yfyx f_x + y f_y:

= \frac{x^2}{\sqrt{x^2 + y^2}} + \frac{y^2}{\sqrt{x^2 + y^2}} = \frac{x^2 + y^2}{\sqrt{x^2 + y^2}} = \sqrt{x^2 + y^2} = f(x, y)$$ Thus, we verify that: $$x f_x + y f_y = f(x, y)$$ ### Conclusion: Using Euler's theorem, we find that $$x f_x + y f_y = \sqrt{x^2 + y^2}$$, which is consistent with the function $$f(x, y)$$. --- Would you like any further details or clarification? Here are 5 related questions: 1. How do you find the degree of homogeneity for a given function? 2. What are the conditions for applying Euler's theorem? 3. Can Euler's theorem be used for non-homogeneous functions? Why or why not? 4. How do you interpret the physical meaning of $$x f_x + y f_y = n f(x, y)$$ for a homogeneous function? 5. How does the method change for functions of three variables (e.g., $$f(x, y, z)$$)? **Tip:** When finding partial derivatives for functions involving square roots, carefully apply the chain rule to avoid errors!

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Math Problem Analysis

Mathematical Concepts

Partial Derivatives
Homogeneous Functions

Formulas

x fₓ + y fᵧ = n f(x, y)
f(x, y) = √(x² + y²)
fₓ = x / √(x² + y²)
fᵧ = y / √(x² + y²)

Theorems

Euler's Theorem on Homogeneous Functions

Suitable Grade Level

Undergraduate Mathematics