By Thomas Szirtes Ph.D P.E.

Utilized Dimensional research and Modeling presents the whole mathematical heritage and step by step tactics for making use of dimensional analyses, besides a variety of purposes to difficulties in engineering and utilized technology, akin to fluid dynamics, warmth movement, electromagnetics, astronomy and economics. This re-creation deals extra worked-out examples in mechanics, physics, geometry, hydrodynamics, and biometry. * Covers four crucial elements and functions: - critical features of dimensional platforms - functions of dimensional thoughts in engineering, arithmetic and geometry - purposes in biosciences, biometry and economics - functions in astronomy and physics* deals greater than 250 worked-out examples and issues of ideas* offers special descriptions of strategies of either dimensional research and dimensional modeling

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Example 1-17 Given matrix A1 = ΄ ΅ 3 3 6 5 5 7 4 7 2 0 . What is its rank? –1 –2 –3 –4 –5 –1 –3 –8 –9 –10 Step 1. Select any nonzero element of A1. Say we select the top left element “3,” which is then marked, as above. We call this element the pivot. Step 2. Generate matrix A2 such that ΄΅ 3 7 · pivot ΄ ΅΄ 0 0 0 0 0 0 –3 –7 – ᎏ239ᎏ – ᎏ335ᎏ 1 3 6 5 5] = A1 – ᎏᎏ · = 0 –1 –1 – ᎏ73ᎏ – ᎏ130ᎏ 3 –3 –3 –6 –5 –5 row of pivot –3 –3 –6 –5 –5 0 –2 –6 – ᎏ232ᎏ – ᎏ235ᎏ ·[3 · 1 A2 = A1 – ᎏᎏ · 3 –1 –1 9 9 18 15 15 21 21 42 35 35 ΅ column of pivot Note that the denominator of the fraction (just before the pivot’s column vector) is the pivot itself (in this case “3”).

Am·b1 am·b2 ··· ··· .. ··· a1·bn a2·bn .. am·bn ΅ (1-12) If vectors a and b contain m and n elements, respectively, then the outer product a·bT is an m × n matrix. Example 1-8 ΄ ΅ ΄ ΅ 2 1 Given column vectors a = –1 and b = 4 , we wish to determine aT·b and a·bT. Thus, 3 –3 ΄ ΅΄ ΅ 2 aT·b = –1 3 T · 1 4 –3 = [2 –1 3]· ΄ ΅ 1 4 –3 = (2)(1) + (–1)(4) + (3)(–3) = –11 and ΄ ΅ 2 a·bT = –1 3 ·[1 4 ΄ 2 8 –6 –3] = –1 –4 3 3 12 –9 ΅ ⇑ 10 APPLIED DIMENSIONAL ANALYSIS AND MODELING Utilizing now the outer products of vectors [see (1-12)], if in A·B, matrix A is partitioned into its columns and B into its rows, then the product obtained is the sum of the outer products, or dyads, formed by the columns of A and the rows of B.

Row of p · ΄΅ · 1 Step 3. Generate matrix An+1 = An – ᎏ p column of p Step 4. If An+1 = 0, then the rank of A1 is n, otherwise set n = n + 1, then go to Step 2. In order to minimize round-off errors, it is usually advisable to select a pivot which has the largest absolute value among the nonzero elements. An alternate approach is to select, if possible, “1” as a pivot. This will obviate the need to deal with fractions. We now introduce the notion of linear dependence. Definition 1-10. Vectors a1, a2, .