ADC 801S
ADVANCED CALCULUS
June 2023
1. (a) If x = rcos0 and y = rsin0, find the (r,0) equations for¢ which obeys Laplace's
equation in two-dimensional caresian co-ordinates
(5)
(b) if Q = log(tanx +tarry+ tanz), show that
.
sm
2x-aau
X
+
.
sm
2y-aauy
+
.
sm
2z-aau
Z
=
2
(5)
(c) If u = x2tan ~, find
a2u I
axay (-1,2)
(5)
2. (a) Minimize J(x1, x2) = X1- X2+ 2xr + 2x1X2+ x~ by taking the starting from the
point X 1 = { ~} using Davidon-Fletcher-Powell (DFP) method with
[Bi] = [~ ~] , E = 0.01
(10)
(b) Minimize J(x1, X2) = X1- X2+ 2xr + 2X1X2+ x~ by taking the starting from the
point X 1 = { ~} , by using Newton's Method
(10)
3. (a) If
show that
r · V¢ = n<p
where n is constant
(8)
(b) Find the directional derivative of the function
cp(x,y, z) = x 2y - 3yz + 2xz
in the direction
n = 4i - 7j + 4k
at the point (3, -2, 1).
(8)
4. (a) Determine the minimum distance between the origin and the hyperbola defined by
x2 + 8xy + 7y2 = 226
(6)
(b) Show that V · (Vgm) = m(m + l)gm- 2, if g =xi+ yj + zk.
(9)
(c) A material body's geometric representation is a planar area R, delimited by the
curves y = x2 and y = - x2 wi~hin the boundaries 0 x 1. The density
function associated with this model is denoted as p = xy.
1