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Assume that all the given functions have continuous second-order partial derivatives.find ∂2

Assume that all the given functions have continuous second-order partial derivatives.find ∂2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Assume that all the given functions have continuous second-order partial derivatives.find ∂2z/(∂r ∂s).

### Understanding Mixed Partial Derivatives In the context of partial differential equations, it's essential to comprehend the second-order partial derivatives and their continuity for given functions. Consider the problem presented below, which delves into finding a mixed second-order partial derivative. **Problem Statement:** Assume that all the given functions have continuous second-order partial derivatives. If \( z = f(x, y) \), where \( x = r^2 + s^2 \) and \( y = 5rs \), find \( \frac{\partial^2 z}{\partial r \partial s} \). (Compare with this [example](#).) ### Solution Approach: To find \(\frac{\partial^2 z}{\partial r \partial s}\), we apply the chain rule for multiple variables. The chain rule allows us to express the partial derivatives of \( z \) with respect to \( r \) and \( s \) in terms of the partial derivatives of \( z \) with respect to \( x \) and \( y \). **Formulation:** \[ \frac{\partial^2 z}{\partial r \partial s} = \left( \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial s} \right) \right) \] Breaking this down with the given transformations \( x = r^2 + s^2 \) and \( y = 5rs \), we use the chain rule to get: \[ \frac{\partial z}{\partial x}, \quad \frac{\partial z}{\partial y}, \quad \frac{\partial^2 z}{\partial x^2}, \quad \frac{\partial^2 z}{\partial y^2}, \quad \text{and} \quad \frac{\partial^2 z}{\partial x \partial y} \] **Mathematical Breakdown:** \[ \frac{\partial^2 z}{\partial r \partial s} = \boxed{\frac{\partial^2 z}{\partial x^2}} + \boxed{\frac{\partial^2 z}{\partial y^2}} + \boxed{\frac{\partial^2 z}{\partial x \partial y}} + \boxed{\frac{\partial z}{\partial y}} \] Here, each box represents a placeholder to be filled in with the appropriate expressions after calculating the derivatives using the given functions
Transcribed Image Text:### Understanding Mixed Partial Derivatives In the context of partial differential equations, it's essential to comprehend the second-order partial derivatives and their continuity for given functions. Consider the problem presented below, which delves into finding a mixed second-order partial derivative. **Problem Statement:** Assume that all the given functions have continuous second-order partial derivatives. If \( z = f(x, y) \), where \( x = r^2 + s^2 \) and \( y = 5rs \), find \( \frac{\partial^2 z}{\partial r \partial s} \). (Compare with this [example](#).) ### Solution Approach: To find \(\frac{\partial^2 z}{\partial r \partial s}\), we apply the chain rule for multiple variables. The chain rule allows us to express the partial derivatives of \( z \) with respect to \( r \) and \( s \) in terms of the partial derivatives of \( z \) with respect to \( x \) and \( y \). **Formulation:** \[ \frac{\partial^2 z}{\partial r \partial s} = \left( \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial s} \right) \right) \] Breaking this down with the given transformations \( x = r^2 + s^2 \) and \( y = 5rs \), we use the chain rule to get: \[ \frac{\partial z}{\partial x}, \quad \frac{\partial z}{\partial y}, \quad \frac{\partial^2 z}{\partial x^2}, \quad \frac{\partial^2 z}{\partial y^2}, \quad \text{and} \quad \frac{\partial^2 z}{\partial x \partial y} \] **Mathematical Breakdown:** \[ \frac{\partial^2 z}{\partial r \partial s} = \boxed{\frac{\partial^2 z}{\partial x^2}} + \boxed{\frac{\partial^2 z}{\partial y^2}} + \boxed{\frac{\partial^2 z}{\partial x \partial y}} + \boxed{\frac{\partial z}{\partial y}} \] Here, each box represents a placeholder to be filled in with the appropriate expressions after calculating the derivatives using the given functions
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