Engineering Analysis I

R
1. Verify that the contour integral C [2xy2 dx+2x2y dy+dz] is independent of the path. Evaluate
this integral between the points (0, 0, 0) and (a, b, c).
2. Given the parametric form of a cone r(u, v) = [u cos v, u sin v, cu] (a) find an explicit representation of the form z =
f(x, y), (b) find and identify the parameter curves defined as u = const
and v = const, and (c) find a normal vector N to the conical surface.

3. In class we discussed surface integrals without regard to orientation. By reparameterizing the
surface integral could be written as
ZZ
ZZ
I=
G(r)dS =
G(r(u, v))|N(u, v)|du dv
S

(a) Consider the case with G = z and the surface S is the hemisphere x2 + y2 + z 2 = 9 with
z ≥ 0. Use polar coordinates and evaluate the right-hand
side of the above result. (b) The
p
surface S is also given explicitly by z = f(x, y) = 9 − x2 − y2 . For such cases the surface
integral can be rewritten as
s
2 2
ZZ
ZZ
∂f
∂f
G(r)dA =
G(x, y, f(x, y)) 1 +
+
dx dy.
∂x
∂y
S
R∗
Evaluate the right-hand side of this result.

RR
ˆ dA using the divergence (Gauss’) theorem when (a) F = [x3 , y3 , z 3 ] and the
4. Evaluate S F•n
surface S is the sphere x2 + y2 + z 2 = 9, and (b) when F = [9x, y cosh2 x, −z sinh2 x] and S is
the ellipsoid 4×2 + y2 + 9z 2 = 36.

5. Consider the vector function F = [ez , ez sin y, ez cos y] and the surface S : z = y2 , 0 ≤ x ≤
4, 0 ≤ y ≤ 2. Stoke’s theorem states that
ZZ
I
ˆ dS =
(∇ × F)•n
F•dr.
S

(a) Evaluate the left-hand side of this result, and (b) evaluate the right-hand side.