Vector Calculus 1
The first rule in understanding vector calculus isdraw lots of pictures. This subject can become rather abstract
if you let it, but try to visualize all the manipulations. Try a lot of special cases and explore them. Keep relating
the manipulations to the underlying pictures and don’t get lost among the infinite series.
9.1 Fluid Flow
When water or any fluid moves through a pipe, what is the relationship between the motion of the fluid and the
total rate of flow through the pipe (volume per time)? Take a rectangular pipe of sidesaandbwith fluid moving
at constant speed through it and with the velocity of the fluid being the same throughout the pipe. It’s a simple
calculation: In time∆tthe fluid moves a distancev∆tdown the pipe. The cross-section of the pipe has area
A=ab, so the volume that move past a given flat surface is∆V =Av∆t. The flow rate is the volume per time,
∆V/∆t=Av. (The usual limit as∆t→ 0 isn’t needed here.)
A
v∆t (a)A
v∆t (b)Just to make the problem look a little more involved I want to know what the result will be if I ask for the
flow through a surface that is tilted at an angle to the velocity. Do the calculation the same way as before, but
use the drawing (b) instead of (a). The fluid still moves a distancev∆t, but the volume that moves past this
flat but tilted surface is not its new (bigger) areaAtimesv∆t. The area of a parallelogram is not the product
of its sides and the volume of a parallelepiped is not the area of a base times the length of another side.
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