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Equilibrium Temperature Distribution
This project gives students an introduction to one type of problem encountered in thermodynamics, that of solving for the equilibrium temperature distribution
of a thin plate of metal. One way of solving this type of problem is to solve a continuous-time differential equation that can be queried as a function of (x,y) for
any continuous-valued position on the plate of metal. In general, this solution could be exact given certain assumptions, but this solution is somewhat difficult
to compute. A simpler way to approximately solve the problem is to discretize the plate and solve a system of linear equations.
Problems
Consider the discretized square plate in Figure 1.
15
X;
A,
5
35
Y,
&
10*
Figure 1: Discretized Square Plate
1. Write a system of linear equations for this system and clearly identify the matrix A in the matrix equation Ax=b.
2. Compute the determinant of A.
3. Compute the classical adjoint matrix (i.e., the adjugate matrix) of A.
4. Using your answers from problems 2 and 3, solve for A-1 and x.
5. Use Cramer's rule to check your answer by solving for just X1 and X4.

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.