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2 Suppose that T (x, y, z) represents the temperature at the point (x, y, z), measured in ƏT Kelvin. What are the units of ax, if "x" is

2 Suppose that T (x, y, z) represents the temperature at the point (x, y, z), measured in ƏT Kelvin. What are the units of ax, if "x" is

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please assist me with question 2 of this homework. Thanks
**Problem Statement** Suppose that \( T(x, y, z) \) represents the temperature at the point \( (x, y, z) \), measured in Kelvin. What are the units of \( \frac{\partial T}{\partial x} \), if \( x \) is measured in miles? What about the units of \( \frac{\partial^2 T}{\partial x^2} \)? **Explanation** The problem involves partial derivatives of temperature with respect to spatial coordinates. Here's a detailed breakdown: - \( T(x, y, z) \): This function represents the temperature at a given point \( (x, y, z) \) in 3-dimensional space. The temperature \( T \) is measured in Kelvin. - \( \frac{\partial T}{\partial x} \): This notation represents the partial derivative of the temperature \( T \) with respect to the spatial coordinate \( x \). The unit of \( \frac{\partial T}{\partial x} \) is determined by the ratio of the unit of temperature to the unit of distance. Since temperature \( T \) is measured in Kelvin (K) and \( x \) is measured in miles (mi), the unit of \( \frac{\partial T}{\partial x} \) is \( \frac{K}{mi} \) (Kelvin per mile). - \( \frac{\partial^2 T}{\partial x^2} \): This notation represents the second partial derivative of the temperature \( T \) with respect to the spatial coordinate \( x \). The unit of \( \frac{\partial^2 T}{\partial x^2} \) is determined by the ratio of the unit of the first partial derivative of temperature to the unit of distance. Since \( \frac{\partial T}{\partial x} \) is measured in \( \frac{K}{mi} \) and distance \( x \) is measured in miles, the unit of \( \frac{\partial^2 T}{\partial x^2} \) is \( \frac{K}{mi^2} \) (Kelvin per square mile). **Summary** - The unit of \( \frac{\partial T}{\partial x} \) is \( \frac{K}{mi} \). - The unit of \( \frac{\partial^2 T}{\partial x^2} \) is \( \frac{
Transcribed Image Text:**Problem Statement** Suppose that \( T(x, y, z) \) represents the temperature at the point \( (x, y, z) \), measured in Kelvin. What are the units of \( \frac{\partial T}{\partial x} \), if \( x \) is measured in miles? What about the units of \( \frac{\partial^2 T}{\partial x^2} \)? **Explanation** The problem involves partial derivatives of temperature with respect to spatial coordinates. Here's a detailed breakdown: - \( T(x, y, z) \): This function represents the temperature at a given point \( (x, y, z) \) in 3-dimensional space. The temperature \( T \) is measured in Kelvin. - \( \frac{\partial T}{\partial x} \): This notation represents the partial derivative of the temperature \( T \) with respect to the spatial coordinate \( x \). The unit of \( \frac{\partial T}{\partial x} \) is determined by the ratio of the unit of temperature to the unit of distance. Since temperature \( T \) is measured in Kelvin (K) and \( x \) is measured in miles (mi), the unit of \( \frac{\partial T}{\partial x} \) is \( \frac{K}{mi} \) (Kelvin per mile). - \( \frac{\partial^2 T}{\partial x^2} \): This notation represents the second partial derivative of the temperature \( T \) with respect to the spatial coordinate \( x \). The unit of \( \frac{\partial^2 T}{\partial x^2} \) is determined by the ratio of the unit of the first partial derivative of temperature to the unit of distance. Since \( \frac{\partial T}{\partial x} \) is measured in \( \frac{K}{mi} \) and distance \( x \) is measured in miles, the unit of \( \frac{\partial^2 T}{\partial x^2} \) is \( \frac{K}{mi^2} \) (Kelvin per square mile). **Summary** - The unit of \( \frac{\partial T}{\partial x} \) is \( \frac{K}{mi} \). - The unit of \( \frac{\partial^2 T}{\partial x^2} \) is \( \frac{
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