13—Vector Calculus 2 401
Example: IfF~=Axyxˆ+B(x^2 +L^2 )yˆ, what is the work done going from point(0,0)to(L,L)along the
three different paths indicated.?
∫
C 1
F~.d~r=
∫
[Fxdx+Fydy] =
∫L
0
dx0 +
∫L
0
dy B 2 L^2 = 2BL^3
∫
C 2
F~.d~r=
∫L
0
dxAx^2 +
∫L
0
dy B(y^2 +L^2 ) =AL^3 /3 + 4BL^3 / 3
∫
C 3
F~.d~r=
∫L
0
dy B(0 +L^2 ) +
∫L
0
dxAxL=BL^3 +AL^3 / 2
3
2 1
Gradient
What is the line integral of a gradient? Recall from section8.5and Eq. (8.10) thatdf= gradf.d~r. The integral
of the gradient is then ∫
2
1
gradf.d~r=
∫
df=f 2 −f 1 (11)
where the indices represent the initial and final points. When you integrate a gradient, you need the function only
at its endpoints. The path doesn’t matter. See problem 19 for a caution.
13.3 Gauss’s Theorem
The original definition of the divergence of a vector field is Eq. (9.9),
div~v= lim
V→ 0
1
V
dV
dt
= lim
V→ 0
1
V
∮
~v.dA~
Fix a surface and evaluate the surface integral of~vover the surface.
∮
S
~v.dA~
dA~
k
k′
ˆnk′ ˆnk