Another Look at Line Integrals

The arc length function s(t) of a curve r(t), t going from a to b, is defined by
s(t)=Integral from a to t of ||r'(t)||

By the Fundamental Theorem of Calculus, s'(t)=||r'(t)||

If you look at a function f(x,y,z) defined along a curve r(t) (for t from a to b) and you consider the curve as being made up of "infinitesimal" pieces of arc lengths of the curve (ds) , the work or flow along the curve (hence the line integral) becomes Integral from a to b of f ds. You often see line integral expressed this way.

If you have a parametrized curve r(t), we reconcile the two approaches in this way:
ds=(ds/dt) dt=||r'(t)|| dt

So f ds is f(x(t),y(t),z(t)) ||r'(t)|| dt

Note that Integral ds is Integral ||r'(t)|| dt which is arc length.

Flow along a curve and work are two different physical ideas but they are represented by the same thing: the line integral.


  f[x_,y_]:=x^2 y


  r[t_]:={3 Cos[t],3 Sin[t]}

The integral of f ds from 0 to Pi is: (let's go piece by piece)


  f[r[t][[1]],r[t][[2]]]


  norm2[D[r[t],t]]


  Simplify[%]


  fds=3 f[r[t][[1]],r[t][[2]]]


  Integrate[81 Cos[t]^2 Sin[t],{t,0,Pi}]

We can do this all at once, of course.


  Integrate[f[r[t][[1]],r[t][[2]]]*norm2[D[r[t],t]],
      {t,0,Pi}]

Up to Some Multivariable Calculus Ideas Part II