ยง16.2 Vector Fields and Line Integrals

Lecture notes

Math3d is good for experimenting with vector fields (click on "Add Object" then "Vector Field")

Vector fields

Line integrals; Work and Circulation

Flux

Examples

  1. Find the work done by \[ \F = -y\i + x\j - z\k \] along
    1. the straight line segments connecting $(0,0,0)\to(1,1,0)\to(1,1,1);$
    2. the curve $\r(t)=(t,t^2,t^3),\quad 0\le t\le 1.$
  2. Find the circulation of
    1. $\F = -y\i + x\j + \k$
    2. $\F = x\i + y\j + z\k$
    around the loop \[ \r(t) = \cos(t)\i + \sin(t)\j\quad 0\le t\le 2\pi. \]
  3. Find the flux of \[ \F = (x-y)\i + (x+y)\j \] across the circle $x^2+y^2=1.$