By H.S. Gangwar, Dr. Prabhakar Gupta

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Y∂x ∂y∂x ∂z ∂z and : ∂ y ∂x Let z = f(x, y) represents the equation of a surface in xyzcoordinate system. Suppose APB is the curve which a plane through any point P on the surface O to the xz-plane, cuts. As point P moves along this curve APB, its coordinates z and x vary while y remains constant. t. x. Geometrical interpretation of Hence, and Z A P O ∂z = tan θ (slope of the curve APB at the point P) ∂x ∂z = tan φ (slope of the curve CPD at point P) ∂y Y f q B D X Fig. 2 22 A TEXTBOOK OF ENGINEERING MATHEMATICS—I Example 1.

If u = f (y – z, z – x, x – y), show that ∂u ∂u ∂u + + = 0. ∂ ∂z y ∂x Sol.

2 2 + a2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 . If log y = tan–1x, show that (1 + x2) yn + 2 + {2(n + 1)x – 1} yn + 1 + n (n + 1) yn = 0 and hence find y3, y4 and y5 at x = 0. U. ), 2003] Ans. y3 (0) = – 1, y4 (0) = – 1, y5 (0) = 5 20 A TEXTBOOK OF ENGINEERING MATHEMATICS—I 4 . If f (x) = tan x, then prove that fn (0) – nc2 fn – 2 (0) + nc4 fn – 4 (0) – ....... = Sin 5 . If y = sin–1x, find yn (0). FG nπ IJ . H 2K Ans. n is odd, yn (0) = (n – 2)2 (n – 4)2 .... 52 . 32 . 1 n is even, yn (0) = 0. 6 . Find yn (0) when y = sin (m sin h–1 x).

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