By H.K. Dass
Offers with partial differentiation, a number of integrals, functionality of a posh variable, exact services, laplace transformation, complicated numbers, and facts.
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Extra info for Advanced Engineering Mathematics
If v be a potential function such that v = v(r) and r = x + y + z , show that 2 v x 2 2 v y 2 2 v z 2 d2v d r2 2 dv r dr 21. Given that w = x + 2y + z2, x = r/s, y = r2 + es, and z = 2r, show that r w w s 12 r 2 2 s e s . r s w when u = 0, v = 0 v If w = (x2 + y – 2)4 + (x – y + 2)3 , x = u – 2 v + 1, and y = 2 u + v – 2 Ans. 99 23. 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.
T. t. t. y, we get u v 0 = 2u 2v y y u v 1 = v y u y On solving (5) and (6), we get v u u v = 2 2, and = 2 2 y y u v u v Example 48. (4) Ans. (6) Ans. y = a (cos u sin v) z = 1 + sin (u – v) z z where p and q mean x and y respectively.. Solution. t. (4) Created with Print2PDF. t. t. t. t. y, we get u v a cos u sin v y y u v = a sin u cos v y y v and y , we get sin v v cos u = , and y a cos u v a cos u v Ans. (9) u Solving (8) and (9) for y u y Putting the values of u , and v in (7), we have y y sin v cos u z q = cos (u – v) y a cos u v a cos u v 1 q = sin v cos u a Ans.
If z(z2 + 3x) + 3y = 0, prove that 2 z z 2 z z 2 z ;z 2 x y xy x y 2z 2 2 z x y 2 x y 2 3. If z = log (e + e ), show that rt – s = 0. 2 z ( x 1) ( z 2 x)3 . 1 1 4. If f (x, y) = x3y – xy3, find f f x y x 1 Ans. 13 22 y 2 5. If = t n e r2 4t , find what value of n will make 1 r 2 r 2 r . r t 6. Show that the function u = arc tan (y/x) satisfies the Laplace equation Ans. n = 2u x 2 2u y 2 3 2 0.