Math 208 (Lutz)
Study Guide: Line Integrals & Flux Integrals
Strategies for computing
R
April 23, 2014
Line Integrals
C
F~ · d~r
1. By looking at the vector field and the path, we can determine whether we expect the line
integral to be positive, negative, or zero. See Section 18.1, p. 928.
R
2. Switching the direction of travel of C changes the sign of C F~ · d~r. See this and other
properties in Section 18.1, p. 931.
3. If F~ is everywhere tangent
to C in the direction of travel of C and has constant magnitude
R
||F~ || = m on C, then
F~ · d~r = m · Length of C. See Section 18.1, Exercise 40, p. 934.
C
4. If ~r(t) is a smooth parameterization of C for a ≤ t ≤ b and F~ is continuous on C, then
Z
F~ · d~r =
Z
b
F~ (~r(t)) · ~r 0 (t) dt.
a
C
See Section 18.2, p. 936.
5. Fundamental Theorem of Calculus for Line Integrals: If C is a piecewise smooth oriented
path starting at P and ending at Q and F~ is a vector field with potential function f , then
Z
F~ · d~r = f (Q) − f (P ).
C
See Section 18.3, p. 941.
Note: If F~ is a continuous vector field that has a potential function, then F~ is path independent.
Conversely, if F~ is path-independent, then F~ has a potential function. See Section 18.3, pp.
943-944.
6. Green’s Theorem: Suppose C is a piecewise smooth simple closed curve that is the boundary
of a region R in the plane, where C is oriented so that R is one the left as we move around
the curve. Suppose F~ = F1~i + F2~j is a smooth vector field on an open region containing R
and C. Then
Z
Z ∂F
∂F
1
2
−
dx dy.
F~ · d~r =
∂x
∂y
C
R
See Section 18.4, p. 952.
Note: If F~ = F1~i + F2~j is path-independent, then
∂F2 ∂F1
−
= 0.
∂x
∂y
The converse holds when, in addition, we the domain of F~ satisfies certain conditions. See Section
18.4, The Curl Test, p. 954.
Note: The curl of a vector field F~ , written curl F~ , measures the circulation of a vector field, i.e.,
the tendency of F~ to cause a paddle-wheel to rotate. It is related to the circulation density. See
Section 20.3, pp. 1010 - 1012.
7. Stokes’ Theorem: If S is a smooth oriented surface with piecewise smooth, oriented boundary
C, and if F~ is a smooth vector field on an open region containing S and C, then
Z
Z
~
~
F · d~r =
curl F~ · dA,
C
S
where the orientation of F is determined from the orientation of S using the right-hand rule.
See Sections 20.3 - 20.4, pp. 1011, 1019.
Flux Integrals
Strategies for Computing
R
S
~
F~ · dA.
~ and F~ is constant (on S), then the flux of F~ through S is
1. If S is flat with area vector A,
Z
~ = F~ · A.
~
F~ · dA
S
See Section 19.1, p. 971.
2. If F~ is everywhere perpendicular to the orientation on S, then the flux of F~ through S is 0.
3. We have techniques for calculating the flux of F~ through S; in each situation we replace the
flux integral with a double integral.
(a) If S is the portion of the graph of z = f (x, y), oriented upward, over the region R in
the xy-plane, then
Z
Z
~
~
F · dA =
F~ (x, y, f (x, y)) · (−fx~i − fy~j + ~k) dA.
S
R
See Section 19.2, p. 980 for details.
(b) If S is a portion of a cylinder of radius R centered on the z-axis, oriented outward, over
the θz-region T , then
Z
Z
~
~
F · dA =
F~ (R, θ, z) · (cosθ~i + sin θ~j + 0~k)R dθ dz.
S
T
See Section 19.2, p. 981 for details.
(c) If S is a portion of a sphere of radius R centered at the origin, oriented outward, over
the θφ-region T , then
Z
Z
~
~
F · dA =
F~ (R, θ, φ) · (sin φ cos θ~i + sin φ sin θ~j + cos φ~k)R2 sin φ dθ dφ.
S
T
See Section 19.2, p. 983 for details.
Note: If you are asked to evaluate a flux integral covered by one of the above three cases,
but the orientation is the opposite, simply evaluate the flux using the above technique and
orientation, then switch the sign of the result.
(d) If S is a portion of a parameterized surface ~r = ~r(s, t) where (s, t) varies in the region
R, then
Z
Z
∂~
r
∂~
r
~=
F~ · dA
F~ (~r(s, t)) ·
×
ds dt,
∂s ∂t
S
R
where the order of the cross product ∂~r/∂s × ∂~r/∂t is chosen to point in the direction
of the orientation on S. See Section 19.3, p 986 for details.
Similarly, the area of the surface S is given by
Z
Z ∂~r ∂~r × ds dt,
1 dA =
∂t S
R ∂s
Note: The divergence of a vector field F~ , written div F~ , measures the outflow per unit volume of
a vector field at a point. See the Geometric and Coordinate Definitions, Section 20.1, p. 996.
4. The Divergence Theorem: If W is a solid region whose boundary S is a piecewise smooth
surface, and if F~ is a smooth vector field on an open region containing W and S, then
Z
Z
~
~
F · dA =
div F~ dV,
S
W
where S is given the outward orientation. See Section 20.2, p. 1005.