# Math 294

# Lecture 2

### Given by Prof. Alan Zehnder on 9/4/95 at 10:10 in Kaufmann Auditorium

# Work Integrals

# 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) = d**r**/dt

Examples:

**F** = xy**i** + y**j** - yz**k**

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** = t**i** + t*t**j** + t*t*t**k**

-- A curved path

In both cases t goes from 0 to 1.

### Summary

To Evaluate the Work Integral on a particle moving along a path C:
- Parametrize the path C.
- Find
**v**(t)
- Subsitute x(t) for x, y(t) for y, etc. into
**F(x,y,t)** to make it a function of t only.(t)
- 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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