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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.

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