Corollary 3 of Euler's Theorem Formula and Proof | Summary and Q&A

TL;DR

Corollary 3 of Euler's Theorem states that if a function u in x and y with degree n has a function z in terms of u that is homogeneous in x and y with degree n, then x d z by d x plus y d z by d y will equal n times z.

Key Insights

• ✖️ Corollary 3 of Euler's Theorem is applicable when a function u in x and y with degree n has a function z in terms of u that is homogeneous in x and y with degree n, providing a relationship between partial derivatives and the function z.
• 🫡 The proof for Corollary 3 involves differentiating two equations obtained from applying the product rule and differentiating the first equation with respect to y.
• ☺️ The result of Corollary 3 simplifies to a partial differential equation involving second derivatives of u with respect to x and y.
• 😄 Corollary 3 can be derived from Corollary 2, which focuses on functions of u that may not be homogeneous.

Transcript

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Questions & Answers

Q: What does Corollary 3 of Euler's Theorem state?

Corollary 3 states that if there is a function u in x and y with degree n and a function z in terms of u that is homogeneous with degree n in x and y, then x d z by d x plus y d z by d y should be equal to n times of z.

Q: How is Corollary 3 related to Corollary 2?

Corollary 2 states that for a function z in terms of u where u may not be homogeneous, we have the equation x times of d u over d x plus y times of d u over d y equal to n times of f of u over f dash of u. Corollary 3 can be derived from Corollary 2 by assuming n times of f of u over f dash of u as g of u.

Q: What is the proof for Corollary 3?

The proof involves differentiating two key equations. The first equation is obtained by applying the product rule to differentiate x d z by d x plus y d z by d y. The second equation is obtained by differentiating the first equation with respect to y. By combining and simplifying these equations, the result provided by Corollary 3 is obtained.

Q: What happens when the partial differential equation is simplified?

Simplifying the partial differential equation leads to x square d 2 u by d x square plus 2 x y d 2 u by d x d y plus y square d 2 u by d y square equal to g dash u minus 1 times of g u, which is the result of Corollary 3.

Summary & Key Takeaways

• Corollary 3 of Euler's Theorem states that if there is a function u in x and y with degree n and a function z in terms of u that is homogeneous with degree n in x and y, then a certain partial differential equation holds.

• The proof for Corollary 3 involves differentiating two key equations, one obtained from applying the product rule and the other obtained from differentiating the first equation with respect to y.

• Finally, combining these equations and simplifying leads to the result provided by Corollary 3 of Euler's Theorem.