By H.K. Dass

Offers with partial differentiation, a number of integrals, functionality of a posh variable, distinct features, laplace transformation, complicated numbers, and data.

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**Example text**

4) [Remember] x y dz is called as the total differential of z. 14 CHANGE OF TWO INDEPENDENT VARIABLES x AND y BY ANY OTHER VARIABLE t. Differentiation of composite function If z = f(x, y) Where x = (t) y = (t) Here z is composite function of t. Dividing (4) by dt , we have Created with Print2PDF. (5) [Remember] dz is called the total differential co-efficient of z. 15 CHANGE IN THE INDEPENDENT VARIABLES x AND y BY OTHER TWO VARIABLES u AND v. Let z = f (x, y) where x = (u, v) y = (u, v) Then from (5), we obtain f z = x u f z = x v and .

If x = u + v + w , y = v w + w u + u v, z = uvw and F is a function of x, y, z, then show that F F F F F F u v w x 2y 3z u v w x y z 24. If u = x + a y and v = x + b y, transform the equation 22. Find 2 2 z x2 5 2 z 2 z 2 z = 0, find the values of a and b. , Summer 2000) Ans. 17 IMPORTANT DEDUCTIONS Let z = f (x, y), then f f dx dy dz = x y If z = 0, dz = 0 f f dx dy 0 = x y f dy x = – dx f y 2 d y We can find by differentiating (1). d x2 f 2 f f q, r, Let = p, y x x2 y p From (1) = .

X y z x u u x4 y4 Example 23. If u = loge , show that x x y y = 3. x y (Nagpur University, Summer 2008, Uttarakhand, I Semester 2008) x4 y 4 u = loge x y Here, u is not a homogeneous function but if Solution. We have, y 4 x 4 1 x4 y 4 x x 3 y z = eu = x y y x x 1 x Then z is a homogeneous function of degree 3. By Euler’s Deduction formula I u u f (u ) eu x y n 3 3 = x y f (u) eu Example 24.