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 of degree , the following relation holds:
where and represent the partial derivatives of with respect to and , respectively. To apply this theorem, we first need to determine whether the given function is homogeneous, and if so, find its degree.
Step 1: Check if is homogeneous
A function is homogeneous of degree if:
For , let's check the behavior when we scale and by :
Thus, , meaning that is a homogeneous function of degree (i.e., ).
Step 2: Apply Euler's theorem
Since is homogeneous of degree 1, Euler's theorem tells us that:
Step 3: Verify by finding and
To confirm, we will compute the partial derivatives of .
Now compute :
= \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!Ask a new question for Free
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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
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