By Tofte M.

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The matrix I has the form 2 1 0 60 1 6 I¼6. 4 .. 0 0 3 ÁÁÁ 0 ÁÁÁ 07 7 7 .. 5 . ÁÁÁ 1 A null matrix (0) is a matrix in which all elements equal 0; thus A0 ¼ 0A ¼ 0. A matrix A is singular if Ax ¼ 0, x = 0 where x is a column vector [Lipschutz, 1968]. Notice that x = 0 if at least one element of x is nonzero. Alternatively, it is possible to show that a matrix A is singular if its determinant is zero. 3 after determinants have been defined.

12 Cartesian coordinate system. 13 33 Circular cylindrical coordinate system. 14. The distance from the origin to point P is the spherical radius rs . The azimuth angle f is measured in the x1 – x2 plane from the positive x1 axis. The polar angle us is measured from the positive x3 axis. 14 Spherical polar coordinate system. 4 0 0 rs , 1 us p 0 f 2p EULER’S EQUATION AND HYPERBOLIC FUNCTIONS A fundamental relation between trigonometric and exponential functions is Euler’s equation, which has the form eia ¼ cos a þ i sin a, i¼ pﬃﬃﬃﬃﬃﬃﬃ À1 (2:4:1) Euler’s equation can be derived by comparing the series expansions of each function in Eq.

13. The perpendicular distance from the x3 axis to point P is the cylindrical radius rc . The angle uc is measured in the x1 –x2 plane from the positive x1 axis. 12 Cartesian coordinate system. 13 33 Circular cylindrical coordinate system. 14. The distance from the origin to point P is the spherical radius rs . The azimuth angle f is measured in the x1 – x2 plane from the positive x1 axis. The polar angle us is measured from the positive x3 axis. 14 Spherical polar coordinate system. 4 0 0 rs , 1 us p 0 f 2p EULER’S EQUATION AND HYPERBOLIC FUNCTIONS A fundamental relation between trigonometric and exponential functions is Euler’s equation, which has the form eia ¼ cos a þ i sin a, i¼ pﬃﬃﬃﬃﬃﬃﬃ À1 (2:4:1) Euler’s equation can be derived by comparing the series expansions of each function in Eq.