Advanced Calculus fi..

92 Advanced Calculus. Fifth EditionHere y = (I.,, . .., y,,), and f is a ver,torfunction (f 1 , . . . , f,,). We then write (2.31 )in the concise form:dy = f, dx.(2.31')Here dx = col (dx,, . . . , d r,), dy = col (dv,. . . . , dy,), and f, 1s an abbreviationfor the Jacobian matrix (af,/a,x,). We can also write ay,/ax, for i)f;/a.r, and arethen led to write (2.3 1') in the formdy = y, dx.which is much like the formula dy = yld.r for functions of one variable.EXAMPLE 1The function f is defined by the equationsHencedy = ['I][ dy2 = 2x12x1 dxi +[2.rr d.r2 - 2x3 d.r3 2.r 2.r2 -2.r3 dxldx - 2x2 dx2 + 2.~dq] = 2.rl -2.r2 :;] kx2].dx3 -2x1 2.rz d.r3d~3 -2.~1 d.rl + 2x2 d.r2 + 2.1.~At the point ( XI. .rz, .x3) = (2, 1, 1) we find (yl, y2. v,) = (4,4.-2) andIf x = (2.01, 1.03, 1.02), then dx = (0.01.0.03,0.02). and the last equation givesso that, approximately, y = (4.06,4.02, -1.94); the exact value is (4.0606.4.0196.- 1.9388).EXAMPLE 2 u = x2 - xy, tr = .x.v + y2. Here the independent variable vector is(x, y), and the dependent variable vector is (14. 21). We haveAt (x, v) = (2, 1). (u, V) = (2.3). and the approximating lincar mapping is