By Xin-She Yang

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**Extra resources for Applied Engineering Mathematics**

**Example text**

Three linearly dependent vectors a, b, c are in the same plane. 3) i=l which is a real number. 1 Vectors When llxll = 1, then it is a unit vector. 6) where a:, {3 are constants. If() is the angle between two vectors a and b, then the dot product can also be written a· b = llall llbll cos(8), 0 ::; () ::; 7r. , () = 1r /2), then we say that these two vectors are orthogonal. Rearranging equation (2. 9) Any vector a in a n-dimensional vector space vn can be written as a combination of a set of n independent basis vectors or orthogonal spanning vectors e1, e2, ...

This will essentially lead to a quasi-steady state. With these assumptions, we have Vie = TiW. Therefore, the angular momentum becomes N h = :Lmirlw = Iw, I= :Lmirl, i=l i=l where I the moment of inertia of the particulate system. The total rotational energy is 1 T= N 2 1 2 1 h2 2 ~miriw = 2Iw = 2T ~ Tmin· l=l In order to minimize T, we have to maximize I because h is constant. v 2 r dm = t foR r 2prdr fo " dO= 21ftp foR r 3dr = 1ftp ~ . 3 Applications Using the density p = mj(t1r R 2 ), we have /disk= 1 2 2mR.

3 Inverse 1\llatrix Algebra is defined as the sum of the diagonal elements, n tr(A) = L~i = a11 + a22 + ... 14) i=l The rank of a matrix A is the number of linearly independent vectors forming the tnatrix. Generally, the rank of A is rank( A) ~ min( m, n). For a n x n square matrix A, it is nonsingular if rank( A)= n. From the basic definitions, it is straightforward to prove the following (AB ... z)T = zT ... 15) IAB .... ZI = IAIIBI ... 19) det(A -1 1 ) = det(Ar det(AB) = det(A)det(B). 21) Inverse The inverse matrix A-t of a square matrix A is defined as A - 1 A = AA - 1 = I.