Mathematical Tools for Physics

9—Vector Calculus 1 256

the motion isn’t likely to be perpendicular to the surface. It’s only thecomponentof the velocity normal to the
surface that contributes to the volume swept out. Useˆnto denote the unit vector perpendicular to∆A, then
this volume is∆Aˆn.~v∆t. This is the same as the the calculation for fluid flow except that I’m interpreting the
picture differently.
If at a particular point on the surface the normalnˆis more or less in the direction of the velocity then
this dot product is positive and the change in volume is positive. If it’s opposite the velocity then the change is
negative. The total change in volume of the whole initial volume is the sum over the entire surface of all these
changes. Divide the surface into a lot of pieces∆Aiwith accompanying unit normalsnˆi, then

∆Vtotal=

∑

i

∆Aiˆni.~vi∆t

Not really. I have to take a limit before this becomes an equality. The limit of this as all the∆Ai→ 0 defines
an integral

∆Vtotal=

∮

dAnˆ.~v∆t

and this integral notation is special; the circle through the integral designates an integral over the whole closed
surface and the direction ofˆnis always taken to be outward. Finally, divide by∆tand take the limit as∆t
approaches zero.
dV
dt

=

∮

dAˆn.~v (8)

The~v.ndAˆ is the rate at which the areadAsweeps out volume as it’s carried with the fluid. Note: There’s
nothing in this calculation that says that I have to take the limit asV → 0 ; it’s a perfectly general expression for
the rate of change of volume in a surface being carried with the fluid. It’s also a completely general expression for
the rate of flow of fluid through a fixed surface as the fluid moves past it. I’m interested in the first interpretation
for now, but the second is just as valid in other contexts.
Again, use the standard notation in which the area vector combines the unit normal and the area: dA~=
ˆndA.

divergence of~v= lim

V→ 0

1

V

dV

dt

= lim

V→ 0

1

V

∮

~v.dA~ (9)

If the fluid is on average moving away from a point then the divergence there is positive. It’s diverging.