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

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Chapter2: Second-order Linear Odes
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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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