8-25. Use line integrals to find the area enclosed by the given curves.
(i) The ellipse, x=acos t y =bsin t, 0 ≤t≤ 2 π.
(ii) The circle, x= cos t y = sin t, 0 < t < 2 π.
(iii) The unit square whose boundaries are x= 0, x = 1, y = 0, y = 1.
8-26. Verify the divergence theorem in the case F =xˆe 1 +yˆe 2 +zˆe 3 and Sis the
surface of the sphere x^2 +y^2 +z^2 =a^2 .Hint: Use spherical coordinates.
8-27. Calculate the flux of the vector field F=zˆe 3 entering and leaving the volume
enclosed by the two spheres x^2 +y^2 +z^2 = 1 and x^2 +y^2 +z^2 = 4 .Does the Gauss
divergence theorem hold for this volume and surface?
8-28. Calculate the flux of the vector field F=yˆe 2 entering and leaving the volume
enclosed by the two cylinders x^2 +y^2 = 1 and x^2 +y^2 = 4,bounded by the planes
z= 0 and z= 2.Does the Gauss divergence theorem hold for this volume and surface?
8-29.
Let Sdenote the surface of a rectangular parallelepiped with unit surface normals
±ˆe 1 ,±ˆe 2 ,±ˆe 3 and write the surface integral
I=
∫∫
S
F·dS=
∫
S 1
F·dS+
∫
S 2
F·dS+
∫
S 3
F·dS+
∫
S 4
F·dS+
∫
S 5
F·dS+
∫
S 6
F ·dS
as a summation of the flux over the six faces of the parallelepiped. Calculate the
above flux integral for F =yˆe 1 +zˆe 2 +xˆe 3
(iii) Consider a unit cube with one vertex at the origin. Calculate the flux entering
or leaving each face of the cube. Sum these fluxes and comment on your result.
8-30. Let F =M(x, y )ˆe 1 +N(x, y )ˆe 2 and use r =xˆe 1 +yˆe 2 to represent the position
of the curve Cand show Green’s theorem in the plane can be represented in either
of the forms
(a)
∫
C
©F·dr =
∫∫
R
(∇× F)·ˆe 3 dxdy or (b)
∫
C
©(F׈e 3 )·ˆends =
∫∫
R
∇· (F׈e 3 ), dxdy