What Is Corollary 1 of Euler's Theorem and Its Proof?

TL;DR

Corollary 1 of Euler's Theorem states that for a homogeneous function of degree n, the equation x² * uₓₓ + 2xy * uₓᵧ + y² * uᵧᵧ = n(n - 1)u holds true. The proof involves using partial differentiation and applies Euler's theorem to show the relationship between the second derivatives and the function's degree.

Transcript

hello in this session we'll see corollary one of euler's theorem we'll see the formula and it's proof so let's say we have a homogeneous function u of degree n and let's say in variable x and y then x square times of second derivative of u with respect to x plus 2 times of x y second derivative of u with respect to x and y and plus y square times o... Read More

Key Insights

• 🤝 Corollary 1 of Euler's Theorem deals with a specific type of homogeneous function.
• ☺️ The formula consists of second derivatives with respect to x and y.
• 🎭 The proof involves applying Euler's Theorem and performing partial differentiations.
• ❓ The resulting formula showcases the relationship between the second derivatives and the degree of the function.
• 🈸 Corollary 1 has important applications in mathematical analysis and problem-solving.
• 🔨 The corollary provides a valuable tool for understanding and manipulating homogeneous functions.
• 🆘 Understanding the proof helps in solving related mathematical problems and equations.

Questions & Answers

Q: What is Corollary 1 of Euler's Theorem?

Corollary 1 of Euler's Theorem deals with a homogeneous function of degree n in variables x and y, providing a formula for its second derivatives.

Q: What is the formula in Corollary 1?

The formula states that x^2 times the second derivative of u with respect to x, plus 2xy times the second derivative of u with respect to x and y, plus y^2 times the second derivative of u with respect to y is equal to n times (n - 1) times u.

Q: How is the proof of Corollary 1 conducted?

The proof involves using Euler's Theorem and partial differentiation to derive equations and manipulate them to arrive at the desired formula, showcasing the relationship between the second derivatives.

Q: What is the significance of Corollary 1?

Corollary 1 provides a general formula and proof for a particular type of homogeneous function, allowing for further analysis and applications in various mathematical contexts.

Summary & Key Takeaways

• The video discusses Corollary 1 of Euler's Theorem, which deals with a homogeneous function of degree n in variables x and y.

• The formula states that x^2 times the second derivative of u with respect to x, plus 2xy times the second derivative of u with respect to x and y, plus y^2 times the second derivative of u with respect to y is equal to n times (n - 1) times u.

• The video provides a step-by-step proof of the corollary, using differentiation and partial derivatives.