Page 272 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 263

10.78. Verify Green’s theorem for a multiply connected region containing two ‘‘holes’’ (see Problem 10.10).

10.79. If Pdx þ Qdy is not an exact differential but ðPdx þ QdyÞ is an exact differential where is some function
of x and y,then is called an integrating factor.(a)Prove that if F and G are functions of x alone, then
ðFy þ GÞ dx þ dy has an integrating factor which is a function of x alone and find . What must be
assumed about F and G?(b)Use (a)to find solutions of the differential equation xy ¼ 2x þ 3y.
0
Ð
3
Ans: ðaÞ ¼ e FðxÞ dx ðbÞ y ¼ cx x, where c is any constant
2
2
2
2
2
2
10.80. Find the surface area of the sphere x þ y þðz aÞ ¼ a contained within the paraboloid z ¼ x þ y .
Ans: 2 a
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
x þ y þ z ,prove that
10.81. If f ðrÞ is a continuously differentiable function of r ¼
ðð ððð
0
r dV
f ðrÞ
f ðrÞ n dS ¼
r
S V
ðð
10.82. Prove that r ð nÞ dS ¼ 0 where is any continuously differentiable scalar function of position and n is
S
a unit outward drawn normal to a closed surface S.(See Problem 10.66.)
10.83. Establish equation (3), Problem 10.32, by using Green’s theorem in the plane.
[Hint: Let the closed region r in the xy plane have boundary C and suppose that under the transformation
x ¼ f ðu; vÞ; y ¼ gðu; vÞ,these are transformed into r and C in the uv plane, respectively. First prove
0
0
ðð ð
that Fðx; yÞ dx dy ¼ Qðx; yÞ dy where @Q=@y ¼ Fðx; yÞ. Then show that apart from sign this last
C
r ð @g @g
integral is equal to Q½ f ðu; vÞ; gðu; vފ du þ dv . Finally, use Green’s theorem to transform this
C 0 @u @v

ðð
@ðx; yÞ
into du dv.
F½ f ðu; vÞ; gðu; vފ

@ðu; vÞ
r 0
10.84. If x ¼ f ðu; v; wÞ; y ¼ gðu; v; wÞ; z ¼ hðu; v; wÞ defines a transformation which maps a region r of xyz space
into a region r of uvw space, prove using Stokes’ theorem that
0
ðð ð ððð
@ðx; y; zÞ
du dv dw
Fðx; y; zÞ dx dy dz ¼ Gðu; v; wÞ

@ðu; v; wÞ
r r 0
where Gðu; v; wÞ F½ f ðu; v; wÞ; gðu; v; wÞ; hðu; v; wފ. State sufficient conditions under which the result
is valid. See Problem 10.83. Alternatively, employ the differential element of volume dV ¼
@r @r @r
du dv dw (recall the geometric meaning).
@u @v @w

10.85. (a) Show that in general the equation r ¼ rðu; vÞ geometrically represents a surface. (b)Discuss the geo-
metric significance of u ¼ c 1 ; v ¼ c 2 , where c 1 and c 2 are constants. (c)Prove that the element of arc length
on this surface is given by
2
2
ds ¼ Edu þ 2Fdu dv þ Gdv 2
@r @r @r @r @r @r
where E ¼ ; F ¼ ; G ¼ :
@u @u @u @v @v @v