CHAP. 6] 6.76.
MULTIPLE, LINE AND SURFACE INTEGRALS. INTEGRAL THEOREMS
(a) Let C be any simple closed curve bounding a region having area A. &i> &2> &s are constants,
179
Prove that if a,, a2, a3(
(5) Under what conditions will the line integral around any path C be zero? 6.77.
Find the area bounded by the hypocycloid x2/a + y2'3 = o2/3. [Hint. Parametric equations are x = a cos3 t, y = a sin3 i, 0 a t S 2ir.]
6.78.
If * = p cos <j>, y = p sin <t>, prove that
6.79.
Verify Green's theorem in the plane f 2
2
p2 <Z0 and interpret. where C is the boundary of the 2
2
region enclosed by the circles x + y = 4 and x + y — 16. 6.80.
(a) Prove that
is independent of the path joining (1,0) •^(1,0)
and (2,1). (6) Evaluate the integral in (a). 6.81.
Evaluate
Zx = iry2
along the parabola
from
(0,0)to(»/2,l). 6.82.
Evaluate the line integral in the preceding problem around a parallelogram with vertices at (0,0), (3,0), (5,2), (2,2).
6.83.
Prove that if x — f ( u , v), y — g(u, v) defines a transformation which maps a region "^ of the xy plane into a region <3f of the uv plane then
by using Green's theorem on the integral
and interpret geometrically.
SURFACE INTEGRALS 6.84.
(a) Evaluate
where S is the surface of the cone z2 = 3(x2 + y2) bounded by z = 0
and z = 3. (6) Interpret physically the result in (a). 6.85.
Determine the surface area of the plane 2x + y + 2z = 16 cut oif by (a) x - 0, y - 0, x - 2, y — 3, (6) * = 0, y = 0 and x2 + y2 = 64.
6.86.
Find the surface area of the paraboloid 2z = x2 + y2 which is outside the cone z =
6.87.
Find the area of the surface of the cone z2 - S(x2 + y2) cut out by the paraboloid z = x2 + y2.
6.88.
Find the surface area of the region common to the intersecting cylinders 2
2
2
x +z = a .
6.89.
x2 + y2 = a2 and
(a) Show that in general the equation r = r(u, v) geometrically represents a surface. (6) Discuss the geometric significance of u = clt v = cz where Cj and c2 are constants, (c) Prove that the element of arc length on this surface is given by ds2 - Edu2 + 2Fdudv + Gdv2 where E =
6.90.
(a) Referring to Problem 6.89, show that the element of surface area is given by dS = (b) Deduce from (a.) that the area of a surface r = T(U, v) is [Hint.
Use the fact that
(A X B) • (C X D) = (A • C)(B • D) - (A • D)(B • C).
du dv.
lu dv. and then use the identity