Advanced Calculus

Advanced Calculus

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 edition. Excerpt: ...of are of such a form as to contain only the variables whose differentials enter. In this case these two may be integrated and the two solutions taken together give the family of curves. Or it may happen that one and only one of these equations can be integrated. Let it be the first and suppose that F(a:, y) = C1 is the integral. By means of this integral the variable x may be eliminated from the second of the equations or the variable y from the third. In the respective cases there arises an equation which may be integrated in the form G(_2/, z, Cl) = C2 or G (ac, z, F) = Cs, and this result taken with F (w, y) = C1 will determine the family of curves. are integrable with the results 1: -1/ = O: 2--22 = G2, and these two integrals constitute the solution. The solution might, of course, appear in very different form; for there are an indefinite number of pairs of equations F(a:, 1/, 2, C1) = 0, G(: i:, 1/, z, C2) = 0 which will intersect in the curves of intersection of 13--1/9 = C1, and 2: '---z = C2. In fact (1/3 + C1) = (z? + C2) is clearly a solution and could replace either of those found above. is the only equation the integral of which can be obtained directly. If y be eliminated by means of this first integral, there results the equation Another method of attack is to use composition and division. X Y Z AX + ).tY-I---vZ Here A, p., v may be chosen as any functions of (ac, y, z). It may be possible so to choose them that the last expression, taken with one of the first three, gives an equation which may be integrated. I/Vith this first integral a second may be obtained as before. Or it may be that two different choices of cents, u., v can be made so as to give the two desired integrals. Oshow more

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  • Paperback
  • 189 x 246 x 8mm | 281g
  • United States
  • English
  • black & white illustrations
  • 1236943422
  • 9781236943422