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MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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49 7 Dimensions To find C , notice that there are six ways to get an abc factor. So C = 6 and then B = 3. Thus ( a + b + c ) 3 = ( a 3 + b 3 + c 3 ) + 3( a 2 b + ab 2 + a 2 c + ac 2 + b 2 c + bc 2 ) + 6 abc . This symmetry solution has several merits. First, it is less prone to mistakes than is multi- plying by brute force. Second, it produces the answer in a meaningful, low-entropy form. The chunks in the solution the terms a 3 + b 3 + c 3 and a 2 b + ab 2 + a 2 c + ac 2 + b 2 c + bc 2 and abc each obey the symmetry that nothing important changes if you permute a , b , and c . Rather than using a brute-force method and then doing hard work to turn the solution into a meaningful form, use symmetry reasoning: Whenever possible, work with quantities that obey the symmetries of the problem. This chapter shows how this idea leads naturally to dimensionless groups, the fundamental idea of dimensional analysis. 7.1 Power of multinationals The first example shows what happens when people take no notice of dimensions. Critics of globalization often make this argument: In Nigeria, a relatively economically strong country, the GDP [gross domestic prod- uct] is \$99 billion. The net worth of Exxon is \$119 billion. ‘When multinationals have a net worth higher than the GDP of the country in which they operate, what kind of power relationship are we talking about?’ asks Laura Morosini. [Source: ‘Impunity for Multinationals’, ATTAC , 11 Sept 2002, [url:nigeria-argument] , retrieved 11 Sept 2006] Before reading further, try to find the most egregious fault in the comparison between Exxon and Nigeria. It’s a competitive field, but one fault stands out. The comparison between Exxon and Nigeria has many problems. First, the comparison exaggerates Exxon’s power by using its worldwide assets (net worth) rather than its as- sets only in Nigeria. On the other hand, Exxon can use its full international power when negotiating with Nigeria, so
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