SPRING 2013 – Calculus 101C – Test 4 – Chapter 14 1A
Use Stokes’s Theorem to find the work done by the force 𝐹 = −𝑦 ! 𝚤 + 𝑧𝚥 + 𝑥𝑘 for a body
moving clockwise (as viewed from above) around a curve C.
C:
the oriented triangle lying in the plane 2𝑥 + 2𝑦 + 𝑧 = 6 as shown in Figure 1.
FIGURE 1
1B
Redo problem 1A but this time find the work by directly evaluating the line integral.
Find the work done by the force 𝐹 = −𝑦 ! 𝚤 + 𝑧𝚥 + 𝑥𝑘 for a body moving clockwise (as viewed
from above) around a curve C.
C:
the oriented triangle lying in the plane 2𝑥 + 2𝑦 + 𝑧 = 6 as shown in Figure 1.
FIGURE 1
2
Use Green’s Theorem to find the work done by the force field
𝐹 = 𝑡𝑎𝑛!! 𝑥 + 𝑦 ! 𝚤 + 𝑒 ! − 𝑥 ! 𝚥
C:
for a body moving around the curve C.
the path enclosing the annular region shown in Figure 2.
FIGURE 2
3
For a force field given by: 𝐹 = 𝑒 ! cos 𝑦 ,
A
Show that
B
Find the potential function.
C
Use the Fundamental Theorem for Line Integrals to calculate the work done by 𝐹 on an object moving
along a curve C from 0 , !! , 1 to 1 , 𝜋 , 3 .
4A
Use Stokes’ Theorem to find the circulation around the closed curve C for the velocity field
𝐹 ∙
!
−𝑒 ! sin 𝑦 ,
2 𝑑𝑟 is independent of path
𝐹 = 𝑦𝑥 𝚤 + −𝑦 𝚥 where C is the half closed circle shown in Figure 3.
FIGURE 3
4B
Redo problem 4A but this time find the circulation by directly evaluating the line integral(s).
Find the circulation around the closed curve C for the velocity field 𝐹 = 𝑦𝑥 𝚤 + −𝑦 𝚥 where C is
the half closed circle shown in Figure 3.
FIGURE 3
5
Let Q be the region bounded by the sphere 𝑥 ! + 𝑦 ! + 𝑧 ! = 1. Use the Divergence Theorem to find
the outward flux of the vector field 𝐹 = 𝑥 ! , 𝑦 ! , 𝑧 ! through the sphere.
6A
Use Gauss’s Theorem to find the outward flux of the vector field 𝐹 = 𝑥 ! 𝑦 , 𝑦 ! across the closed
curve C.
C:
the unit square oriented counterclockwise as shown in Figure 4.
FIGURE 4
6B
Redo problem 6A but this time find the outward flux by directly evaluating the line integral(s).
Find the outward flux of the vector field 𝐹 = 𝑥 ! 𝑦 , 𝑦 ! across the closed curve C.
C:
the unit square oriented counterclockwise as shown in Figure 4.
FIGURE 4