Question

A thin plate has a temperature distribution of \(T(x, y)=x^{2}-y^{3}-x^{2} y+\) \(y+20\) for \(0 \leq x, y, \leq 2\). Find the coldest and hottest points on the plate.

Short answer

The coldest point on the plate is (2,2) with a temperature of 12 units and the hottest point is (1, -sqrt(2)) with a temperature of 24.59 units.

Step-by-step solution

Compute the partial derivatives

First, compute the partial derivatives of temperature distribution function with respect to \(x\) and \(y\). The derivative of \(T\) with respect to \(x\), \(T_{x}\) is \(2x-y^{2}\) and with respect to \(y\), \(T_{y}\) is \(-3y^{2}-x^{2}+1\).

Find the critical points

The critical points occur when \(T_{x} = 0\) and \(T_{y} = 0\). Solving these equations, we arrive at the points (0, 0), (0, 1), (0, -1), (1, \(\sqrt{2}\)), (1, -\(\sqrt{2}\)).

Evaluate the temperature at the critical points

Next, we substitute the critical points into the temperature function. Doing this, we find that \(T(0,0) = 20\), \(T(0,1) = 20\), \(T(0,-1) = 22\), \(T(1, \sqrt{2}) = 19.59\), \(T(1, -\sqrt{2}) = 24.59\).

Evaluate the temperature at the boundaries

The edge values are also possible extremums. On x = 0, 2 and y = 0, 2, evaluating the function gives us \(T(2,0) = 16\), \(T(2,2) = 12\), \(T(0,2) = 16\) and \(T(0,0) = 20\).

Compare all the evaluated temperatures

Comparing all the temperatures, the maximum temperature is 24.59 at point (1,-sqrt(2)) and the minimum temperature is 12 at point (2,2).

Key concepts

Temperature Distribution

Understanding how temperature varies across different points on a surface is crucial in many fields ranging from engineering to meteorology. Temperature distribution refers to the spatial variation in temperature across a body or region. In our exercise, the temperature distribution over a thin plate is given by the mathematical function \(T(x, y)=x^{2}-y^{3}-x^{2}y+y+20\). This function represents how temperature changes with the coordinates \(x\) and \(y\) on the plate.

To analyze this distribution, we employ partial derivatives which show how the temperature changes with respect to one variable while keeping the other constant. By setting these derivatives to zero, we can locate critical points which may correspond to the hottest or coldest spots on the plate. This technique is of paramount importance in optimizing systems for heat diffusion or in identifying areas that might be structurally vulnerable due to temperature extremes.

Critical Points Analysis

In mathematics, critical points are locations on a graph where the function's derivative is zero or undefined. In the context of our temperature distribution function, these are points where the temperature does not change locally with minor movements in any direction on the plate.

Identifying Critical Points

In our exercise, the critical points are found by setting the partial derivatives of \(T\) with respect to \(x\) and \(y\) to zero. These points potentially represent local maxima or minima in temperature, which, for practical purposes, could indicate the hottest and coldest spots. The solution methodology is a systematic approach to finding these points by solving \(T_x = 0\) and \(T_y = 0\) simultaneously before evaluating the original temperature function at these critical points.

Evaluating Extrema

The nature of these critical points (whether they are maximum, minimum, or saddle points) can be assessed by further tests such as the second derivative test or by evaluating the function at boundaries if the domain is restricted, a step which should not be overlooked as done in the initial problem-solving steps provided.

Mathematical Methods for Physicists

The field of physics often relies on mathematical tools to model and solve complex problems, from quantum mechanics to thermodynamics. The mathematical methods used include algebra, calculus, and differential equations, to name a few.

Role of Calculus in Physics

Calculus, particularly the study of derivatives and integrals, is one of the primary tools physicists use to analyze the behavior of physical systems. In our temperature distribution problem, calculus helped identify critical points and temperature variations across the plate’s surface by applying the concept of partial derivatives.

Physics and Critical Points

Critical points analysis is a valuable method in physics when examining equilibrium points in a system, or when determining stability. Although in our problem we analyzed the temperature distribution, the same mathematical principles could be applied to other physical scenarios, such as finding the potential energy minimum in a gravitational field or analyzing the hydrodynamic flow around an obstacle. These mathematical methods are not only tools for solving problems but also provide a deeper understanding of physical phenomena.