TAILIEUCHUNG - Ebook Mathematics for engineers (4/E): Part 2

Part 2 book “Mathematics for engineers” has contents: Using matrices and determinants to solve equations, techniques and applications of differentiation, applications of integration, differential equations, the laplace transform, and other contents. | Using matrices and determinants to solve equations Chapter 13 This chapter looks at some of the ways in which matrices and determinants can be used to solve linear simultaneous equations. Block 1 introduces Cramer’s rule, which expresses the solution to simultaneous equations as a ratio of two determinants. The next block shows how simultaneous equations may be written in matrix form. Once in this form an inverse matrix method can be used to solve them. Block 3 explains the method of Gaussian elimination. When the simultaneous equations have been written as a matrix, the rows of the matrix are systematically manipulated to yield the solution. The block includes an explanation of the way in which the manipulation is carried out. Matrices are sometimes used in calculations that model vibrating objects, and in particular those vibrating because of forces in stretched and compressed springs. In such applications engineers are interested in the possible frequencies of vibration. These frequencies are related to what are known as the eigenvalues of a matrix. The eigenvectors represent the various modes of vibration. Block 4 will explain how the eigenvalues and eigenvectors of a matrix are calculated. As they are usually studied in terms of linear systems, they are included here. In many practical applications, approximate solutions of linear equations are obtained using what are called numerical methods. These are introduced in Block 5. The final block focuses on the application of matrices to the analysis of electrical networks. Engineers often need to know the currents in the various branches of a circuit. The idea of branch currents is introduced. A set of simultaneous equations can be formed that models these branch currents, and which can be solved using matrix methods. Chapter 13 contents Block 1 Cramer’s rule Block 2 Using the inverse matrix to solve simultaneous equations Block .

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