Symmetry of second partial derivatives

TL;DR

Learn how to calculate second partial derivatives of multivariable functions and how Schwarz's theorem proves that the order of partial derivatives does not matter.

Transcript

• [Voiceover] So in the last couple videos, I talked about partial derivatives of multivariable functions. And here, I want to talk about second partial derivatives. So I'm gonna write some kind of multivariable function. Let's say it's well, sine of x times y squared. Sine of x multiplied by y squared. And if you take the partial derivative, you h... Read More

Key Insights

• 🫡 Partial derivatives of multivariable functions can be calculated by treating one variable as a constant and differentiating with respect to the other.
• 🥡 Second partial derivatives involve taking the partial derivative of the first partial derivatives.
• 🪈 Schwarz's theorem states that the order of partial derivatives does not matter if the second partial derivatives are continuous.
• ☺️ The notations for second partial derivatives can be written as "partial squared f divided by partial x squared" or as "partial x, x" for convenience.
• 🥹 Continuity of second partial derivatives is a criterion for Schwarz's theorem to hold.
• 🖐️ Playing around with different multivariable functions can help understand and apply Schwarz's theorem.

Questions & Answers

Q: What are second partial derivatives?

Second partial derivatives involve taking the partial derivative of the first partial derivatives of a multivariable function.

Q: What happens if you take the partial derivative with respect to x first and then with respect to y?

The partial derivative with respect to x first and then with respect to y gives the same result as taking the partial derivative with respect to y first and then with respect to x, provided the second partial derivatives are continuous.

Q: Why is Schwarz's theorem important?

Schwarz's theorem is important because it proves that the order of partial derivatives does not matter if the second partial derivatives are continuous, simplifying the calculations for multivariable functions.

Q: Are all functions subject to Schwarz's theorem?

No, Schwarz's theorem applies only to functions where the second partial derivatives are continuous at the relevant point.

Summary & Key Takeaways

• Second partial derivatives are obtained by taking the partial derivative of the first partial derivatives of a multivariable function.

• The order of partial derivatives does not matter, and both paths lead to the same result if the second partial derivatives of the function are continuous.

• Schwarz's theorem states that if the second partial derivatives are continuous, then the order of partial derivatives is interchangeable.