What is the application of Euler's theorem for the given function and how do you differentiate it?
Understand the Problem
The question is asking to apply Euler's theorem to differentiate a given homogeneous function of two variables, and to solve for specific cases provided in the examples.
Answer
For the example \( z = (x^2 + y^2)^{1/3} \), the result is \( -\frac{2}{9}z \).
Answer for screen readers
For the first case, applying Euler's theorem gives: [ = -\frac{2}{9}z ]
Steps to Solve
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Identify the Homogeneous Function We need to confirm that ( z ) is homogeneous of degree ( n ). For example, if ( z = (x^2 + y^2)^{1/3} ), we can find its degree.
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Determine Degree of Homogeneity The degree ( n ) can be obtained by analyzing the function. For ( z = (x^2 + y^2)^{1/3} ): [ n = \frac{1}{3} ] Since it is not an integer, we should apply the theorem accordingly.
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Apply Euler's Theorem Use Euler’s theorem formula: [ x^2\frac{\partial z}{\partial x} + 2xy\frac{\partial z}{\partial y} + y^2\frac{\partial z}{\partial y^2} = n(n-1)z ]
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Differentiate ( z ) with Respect to ( x ) and ( y ) Calculate the necessary partial derivatives:
- ( \frac{\partial z}{\partial x} )
- ( \frac{\partial z}{\partial y} )
For ( z = (x^2 + y^2)^{1/3} ): [ \frac{\partial z}{\partial x} = \frac{2/3 \cdot x}{(x^2 + y^2)^{2/3}} ] [ \frac{\partial z}{\partial y} = \frac{2/3 \cdot y}{(x^2 + y^2)^{2/3}} ]
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Substitute the Derivatives into the Formula Plugging these values into the Euler's theorem equation: [ x^2 \cdot \frac{2}{3} \cdot \frac{x}{(x^2 + y^2)^{2/3}} + 2xy \cdot \frac{2}{3} \cdot \frac{y}{(x^2 + y^2)^{2/3}} + y^2 \cdot 0 = n(n-1)z ]
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Simplify and Solve Combine the terms and simplify to find the expression.
For the other examples provided in the document, follow similar steps to confirm and differentiate each function.
For the first case, applying Euler's theorem gives: [ = -\frac{2}{9}z ]
More Information
Euler's theorem relates the derivatives of a homogeneous function to the function itself. This theorem can simplify complex differentiation tasks and helps verify relationships between variables.
Tips
- Forgetting to check if the function is homogeneous of degree ( n ).
- Miscalculating partial derivatives, especially for complicated functions.
- Not simplifying the results correctly when substituting back into the equation.
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