Problem 69
Question
Suppose a thin metal plate occupies the first quadrant of the \(x y\) plane and the temperature at \((x, y)\) is given by $$ T(x, y)=x y $$ Describe the isothermal curves, that is, the level curves of \(T\).
Step-by-Step Solution
Verified Answer
The isothermal curves are hyperbolas given by \( xy = c \) in the first quadrant.
1Step 1: Understanding Isothermal Curves
Isothermal curves are lines where the temperature is constant. For the given function \( T(x, y) = xy \), these will be curves where \( xy = c \), with \( c \) being a constant.
2Step 2: Expressing Isothermal Curves
Given the equation for temperature \( T(x, y) = xy \), the isothermal curves are given by \( xy = c \). This equation represents a family of hyperbolas in the first quadrant where \( c \) is any positive constant value.
3Step 3: Analyzing the Curve
In the first quadrant, valid isothermal curves arise from the equation \( xy = c \) using positive \( x \) and \( y \) values. Each value of \( c > 0 \) corresponds to a different hyperbola.
4Step 4: General Form of the Curve
The equation \( xy = c \) can be rewritten in standard hyperbolic form as \( y = \frac{c}{x} \). This form makes it clear that for each \( c \), \( y \) relates inversely with \( x \), creating a hyperbola asymptotic to both the \( x \) and \( y \) axes.
Key Concepts
Level CurvesHyperbolasTemperature Distribution
Level Curves
Level curves, also known as contour lines, are essential in representing functions of two variables. These curves are drawn on a plane where the function values are constant. In the context of our exercise, the temperature on a thin metal plate depends on two variables, described by the equation \( T(x, y) = xy \). The level curves for this function are the isothermal curves, where the temperature remains consistent across each curve.
For our specific function, the level curves are defined by the equation \( xy = c \), where \( c \) is a constant. These curves represent the places on the metal plate where the temperature is equal to \( c \). The significance of these curves lies in their ability to visually demonstrate how the temperature varies across the plate. By analyzing the level curves, students can observe patterns in the temperature distribution.
For our specific function, the level curves are defined by the equation \( xy = c \), where \( c \) is a constant. These curves represent the places on the metal plate where the temperature is equal to \( c \). The significance of these curves lies in their ability to visually demonstrate how the temperature varies across the plate. By analyzing the level curves, students can observe patterns in the temperature distribution.
Hyperbolas
Our level curves, expressed by the equation \( xy = c \), form a family of curves known as hyperbolas. Each hyperbola corresponds to a specific value of \( c \), indicating different temperature levels. A hyperbola is a type of conic section and looks somewhat like two mirrored parabolas.
- In the equation \( xy = c \), we see that \( y = \frac{c}{x} \), which is a classic shape of a hyperbola in the plane.
- Each hyperbola bends toward the positive \( x \) and \( y \) axes but never touches them. Thus, these axes are the asymptotes of the hyperbolas.
It's easy to visualize: as you increase or decrease \( c \), the hyperbola shifts closer to or further from the axes, representing different constant temperature values on the plate.
- In the equation \( xy = c \), we see that \( y = \frac{c}{x} \), which is a classic shape of a hyperbola in the plane.
- Each hyperbola bends toward the positive \( x \) and \( y \) axes but never touches them. Thus, these axes are the asymptotes of the hyperbolas.
It's easy to visualize: as you increase or decrease \( c \), the hyperbola shifts closer to or further from the axes, representing different constant temperature values on the plate.
Temperature Distribution
The equation \( T(x, y) = xy \) gives us a clear view of how temperature spreads over the metal plate. Each value of \( (x, y) \) yields a specific temperature, which is an important concept in heat distribution and thermodynamics.
- The isothermal curves show regions of constant temperature. They are useful to map how heat is spread across the plate.
- By looking at these curves, you can infer where the temperature is higher or lower based on the density and arrangement of these curves.
The interesting part becomes apparent when you realize that knowledge of these distributions helps predict thermal behavior, essential in various engineering applications. The inverse relationship between \( x \) and \( y \) along the hyperbolas indicates the dynamic interaction between different regions of the plate, influencing how the temperature changes spatially. This makes these curves incredibly useful for visualizing and understanding temperature behaviors in applied physics scenarios.
- The isothermal curves show regions of constant temperature. They are useful to map how heat is spread across the plate.
- By looking at these curves, you can infer where the temperature is higher or lower based on the density and arrangement of these curves.
The interesting part becomes apparent when you realize that knowledge of these distributions helps predict thermal behavior, essential in various engineering applications. The inverse relationship between \( x \) and \( y \) along the hyperbolas indicates the dynamic interaction between different regions of the plate, influencing how the temperature changes spatially. This makes these curves incredibly useful for visualizing and understanding temperature behaviors in applied physics scenarios.
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