Reference no: EM132723834

Question 1: Sketch the graph of the level surface f (x, y, z) = c at the given value of c

f(x, y) = -4x/(x2+y2+1) = c, c =1

Question 2:Find the total different coefficient of x2y w.r.t x, where x and y are connected by the relation x2 + xy + y2 =1

Question 3. If, u3 + v3 + w3 = x + y + z, u2 + v2 + w2 = x3 + y3 +z3, u + v+ w = x2 + y2 + z2,

then. show that ∂(u, v, w)/∂(x, y, z) = ( x - y)(y - z) (z - x)/ (u -v)(v -w)(w - u)

Question 4. If u = x2/a3 + y2/b3 + z2/c3, where x+y+z = 1, then prove that the stationary value of a is given by

x = a3/(a3=\+b3+c3), y=b3/(a3+b3+c3), z= c3/a3+b3+c3

5. If y1 = X2X3/x1, y2 = X1X3/x2 y3 = x2x1/x3 prove that ∂(y1, y2, y3)/∂(x1, x2, x3) =4

6. Find the maximum and minimum values of x3 + y3 -3axy

7. Show that Γ(1/4)Γ(3/4) = 2 ₀∫^Π/2√tanxdx = Π√2

8. Change the order of integration ₀∫1 √(1-x)∫2+x f(x,y)dydx

9. Express as a single integral and evaluate ₀∫1 ₀∫y dydx + ₂∫4 ₀∫4-x dydx

10. Show that log(1+ x + x2 +x3 ) = x + x2/2 + x3/3 - 3/4 x4 + 1/5 x6 + 1/7 x7 - 3/8 x8

11. Show that ₀∫∞ xn-1 e-ax cosbxdx = Γ(n)/(a2+b2)n/2 cos(n tan-1(b/a))

12. Change the order of integration and evaluate ₀∫a √ax∫a y2/√(y4 -a2x2).dxdy

13 Find the constants a and b so that the surface 2ax² -3byz = (a +2)x will be orthogonal to the surface 4x2 y + z3 = 4 at the point (1, -1, 2).

14. Find the rate of change of Φ = xy² + yz3 at (1, 1,-1) along the direction 2i + j - 2k.

15. Show that i) ∇.(rnr-) = (n + 3)rn

ii) ∇ x (rnr-) = 0-, where r- = xi +yj + zk^ and r = |r-|

16. Prove that f- =(x2 -yz) i^ + (y2 -zx)j^ + (z2 - xy)k^ is
i) Conservative

ii) Find the scalar potential of f-.

iii) find the work done in moving the object from P(0, 1, -1) to Q(Π/2, -1, 2)

17. ∫∫_s∇ x f- n^ ds by using Stoke's theorm, where s is the surface of the Paraboloid x2 + y2 = 2z and whose bounding curve is x2 + y2 = 4, z = 2 and f- =  3yi^ -xzj^ yz²k

18. Verity divergence theorem for F = xi + yj + zk over the volume of the sphere x2 + y2 + z2 = a2.

19. Using Stokes theorem and divergence theorem prove that CurlgradΦ = 0- and DivcurlF- = 0.

20. Verify the Stokes theorem for F- = x2i + xyj, where C is the boundary of the rectangle x=0,y=0, x=1,y=1.