Mathematical Tools for Physics

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