# Vector Fields

In Lecture 1 we learned how to evaluate a line integral, and did a short, fairly simple example. In this lecture we discuss an application of the line integral - computing the work done to move an object through a Force field.

• Vector Field - A vector valued quantity defined over some space.

For example - The Flow Field for a Fluid, which gives the velocity of the fluid at every point in a given space.
Another Example would be the heat flow vector:

In General, a Vector Field F, will be defined by

F(x,y,z) = M(x,y,z)i + N(x,y,z)j + P(x,y,z)k

#### Line Integrals Involving Vector Fields

##### The Work Integral

Given a path and a force field:

F = Vector Force Field Acting on Particle
T = Unit Tangent Vector
n = Unit Normal Vector

The work increment in moving a differential length ds along the path is
dW = F (dot) T ds
= Force dotted with direction, times distance.

The total work done by force on a particle as it moves along a path C is:

This is a line integral, with f(x,y,z) = F (dot) T

So, we just define C: r(t) = x(t)i + y(t)j + z(t)k a<=t<=b
ds = ||v(t)|| dt
T = v(t)/||v(t)||
v(t) = dr/dt

Examples:
F = xyi + yj - yzk
Find the work required to move a particle from (0,0,0) to (1,1,1) along two different paths
C1: r = t(i + j + k) -- A straight line
C2: r = ti + t*tj + t*t*tk
-- A curved path
In both cases t goes from 0 to 1.

• Work along C1: F(x,y,z) = t*ti + tj - t*tk
v = i + j + k

• Work along C2
F(x,y,z) = t^3i + t^2j - t^5k
v = i + 2tj + 3t^2k

Note that work here was different depending on the path chosen for the particle.

### Summary

To Evaluate the Work Integral on a particle moving along a path C:
1. Parametrize the path C.
2. Find v(t)
3. Subsitute x(t) for x, y(t) for y, etc. into F(x,y,t) to make it a function of t only.(t)
4. Do the Integral from a->b of F( x(t), y(t), z(t) ) (dot) v(t) dt

Lecture written by Lawrence C. Weintraub on Sunday, September 10, 1995
Edited by Aric Shafran on Sunday, September 10, 1995
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