Question

Let \(T(x, y)\) be the temperature at a point \((x, y)\) in the plane. Draw the isothermal curves corresponding to \(T=\frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 0\) if $$ T(x, y)=\frac{x^{2}}{x^{2}+y^{2}} $$

Short answer

Isothermal curves: \(y = \pm 3x\), \(y = \pm 2x\), \(y = \pm x\). Curve for \(T=0\) is undefined in finite terms.

Step-by-step solution

Understand the Function

The function given is \( T(x, y) = \frac{x^{2}}{x^{2}+y^{2}} \). This is a rational function where the temperature at any point \((x, y)\) is determined by the ratio of \(x^2\) to \((x^2 + y^2)\). We are interested in plotting isothermal curves, which are curves where the temperature \(T\) is constant.

Set Up Equations for Isothermal Curves

For isothermal curves where \( T(x, y) = C \), rewrite the equation as \( C = \frac{x^{2}}{x^{2}+y^{2}} \). Rearrange it to express \( y^2 \) in terms of \( x^2 \): multiply both sides by \( x^2 + y^2 \) to get \( C(x^2 + y^2) = x^2 \). Simplify to \( Cx^2 + Cy^2 = x^2 \), leading to \( Cy^2 = x^2(1-C) \).

Solve for \( y^2 \)

From the equation \( Cy^2 = x^2(1-C) \), divide both sides by \( C \) to solve for \( y^2 \), resulting in \( y^2 = \frac{x^2(1-C)}{C} \). This shows the relationship between \( x^2 \) and \( y^2 \) for a given \( C \).

Solve for Specific Values of \(C\)

Evaluate the expression \( y^2 = \frac{x^2(1-C)}{C} \) for each given value of \(C\):- For \( C = \frac{1}{10} \), \( y^2 = \frac{x^2(1-\frac{1}{10})}{\frac{1}{10}} = 9x^2 \).- For \( C = \frac{1}{5} \), \( y^2 = \frac{x^2(1-\frac{1}{5})}{\frac{1}{5}} = 4x^2 \).- For \( C = \frac{1}{2} \), \( y^2 = \frac{x^2(1-\frac{1}{2})}{\frac{1}{2}} = x^2 \).- For \( C = 0 \), \( y^2 = \frac{x^2(1-0)}{0} \), which leads to an undefined curve since \(C\) cannot be zero in the denominator. A trivial point occurs where \(y\) is undefined when \(x > 0\), or \( y = \infty \).

Draw the Isothermal Curves

- **For \( C = \frac{1}{10} \):** The curve is \( y = \pm 3x \), a pair of lines passing through the origin with a steeper slope.- **For \( C = \frac{1}{5} \):** The curve is \( y = \pm 2x \), another pair of lines passing through the origin with a slope of 2.- **For \( C = \frac{1}{2} \):** The curve is \( y = \pm x \), representing the lines \( y = x \) and \( y = -x \).- **For \( C = 0 \):** There is no well-defined curve in finite \((x,y)\)-plane, but conceptually, \( (x, y) \, can \, take \, (0, \infty)\, or \infty in \) limits.

Key concepts

Isothermal Curves

Isothermal curves are fascinating concepts in multivariable calculus, representing points on a plane where a temperature function remains constant. Imagine these curves as lines drawn on a map forming areas of the same temperature. These are crucial for understanding how temperature varies in a spatial environment.

Let's consider the example where at each point (x, y) on a plane, the temperature is given by a rational function. To find isothermal curves for specific temperatures, substitute distinct values, like \(T=\frac{1}{10}, \frac{1}{5}, \frac{1}{2}, 0\), into the temperature function. Set the function equal to these values and solve for the relationship between \(x\) and \(y\). Each result outlines an isothermal curve.

This practical application helps us visualize and study how temperature remains uniform along particular paths, letting us predict and analyze thermal behaviors, essential for fields like meteorology, engineering, and environmental science.

Temperature Function

A temperature function describes how temperature varies across a surface or through space. In our problem, the temperature is modeled by the rational function \(T(x, y) = \frac{x^2}{x^2+y^2}\). Such a function is "rational" because it is expressed as the quotient of two polynomials.

Here, \(x^2\) is divided by \(x^2 + y^2\), defining how temperature is distributed at any point \((x, y)\) in the plane. When studying isothermal curves, we focus on scenarios where the temperature is constant. We equate this function to constant values to find the curves where this happens.

This approach is powerful in physics and engineering, helping predict temperature changes in materials and fluids. By carefully analyzing the temperature function, we can control and optimize conditions in various applications.

Rational Function

Rational functions are mathematical expressions of the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. In our exercise, the function \(T(x,y) = \frac{x^2}{x^2 + y^2}\) is a classic example of a rational function. These functions often exhibit interesting behaviors, including asymptotes, where the function tends to infinity, and are key in calculus due to their diverse applicability.

In this context, the numerator \(x^2\) and the denominator \(x^2 + y^2\) determine the temperature at any given point \((x, y)\). As \(x\) or \(y\) approach zero or infinity, the behavior of \(T(x, y)\) changes, offering insights into how temperature varies across the plane. This is pivotal in the analysis of physical systems where variables interact through reciprocal relationships.

Plane Geometry

Plane geometry is the field of mathematics that concerns itself with the shapes, figures, and configurations in a two-dimensional space. In this exercise, we visualize the temperature distribution and isothermal curves across a flat plane. Understanding these geometries aids in interpreting and predicting phenomena in various sciences, including physics and geography.

As derived, the isothermal curves for different temperatures are lines or undefined curves on this plane. We identify these geometric lines using equations like \(y = nx\), where \(n\) is a constant resulting from solving the temperature equations. For example, the lines \(y = 3x\), \(y = 2x\), and \(y = x\) offer visual representations of this geometric interpretation.

This foundational understanding helps create settings for complex problem-solving in fields like architecture and engineering, promoting practical designs and solutions.