Math Problem Statement
Verify Euler's theorem for the function ( u=x^{2}+y^{2}+2 x y )
Solution
Euler's theorem states that for a homogeneous function of degree , the following relationship holds:
First, let's verify if is a homogeneous function and determine its degree.
A function is homogeneous of degree if:
Let's test :
This shows that , so is a homogeneous function of degree 2.
Next, we compute the partial derivatives of :
Now, we check Euler's theorem:
Substitute the partial derivatives:
Notice that:
So,
Therefore, Euler's theorem is verified for the function with .
Would you like further details or have any questions?
Here are 8 related questions to expand your understanding:
- What is a homogeneous function in general terms?
- How can Euler's theorem be applied to functions of more than two variables?
- Can Euler's theorem be used for non-homogeneous functions?
- What is the geometric interpretation of homogeneous functions?
- How does the degree of a homogeneous function affect its properties?
- Can you provide an example of a homogeneous function of degree 3?
- How would you verify Euler's theorem for the function ?
- How does Euler's theorem relate to the concept of scaling in physics and engineering?
Tip: When verifying Euler's theorem, always start by confirming the function's degree of homogeneity before applying the partial derivative operations.
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Math Problem Analysis
Mathematical Concepts
Homogeneous functions
Partial derivatives
Euler's theorem
Formulas
Euler's theorem: x * du/dx + y * du/dy = n * u
Theorems
Euler's theorem for homogeneous functions
Suitable Grade Level
Advanced undergraduate level
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