Posted: November 4th, 2015

Mathematics

Mathematics

1. Verify that the contour integral R
C
[2xy2 dx+2x
2y 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
I =
Z Z
S
G(r)dS =
Z Z
R
G(r(u, v))|N(u, v)|du dv
(a) Consider the case with G = z and the surface S is the hemisphere x
2 + y
2 + z
2 = 9 with
z = 0. Use polar coordinates and evaluate the right-hand side of the above result. (b) The
surface S is also given explicitly by z = f(x, y) = p
9 – x
2 – y
2. For such cases the surface
integral can be rewritten as
ZZ
S
G(r)dA =
Z Z
R*
G(x, y, f(x, y))s
1 + 
?f
?x2
+

?f
?y 2
dx dy.
Evaluate the right-hand side of this result.
4. Evaluate RR
S F•nˆ dA using the divergence (Gauss’) theorem when (a) F = [x
3
, y3
, z3
] and the
surface S is the sphere x
2 + y
2 + z
2 = 9, and (b) when F = [9x, y cosh2
x, -z sinh2
x] and S is
the ellipsoid 4x
2 + y
2 + 9z
2 = 36.
5. Consider the vector function F = [e
z
, ez
sin y, ez
cos y] and the surface S : z = y
2
, 0 = x =
4, 0 = y = 2. Stoke’s theorem states that
Z Z
S
(? × F)•nˆ dS =
I
C
F•dr.
(a) Evaluate the left-hand side of this result, and (b) evaluate the right-hand side.

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