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Chapter 14: Q 20. (page 1106)

Use your intuition from Exercise 19 to propose a formula for the average value of $F (x, y, z)$ along a smooth curve $C$ parametrized by $r (t)$ for $a 鈮� t 鈮� b$ .

Short Answer

Expert verified

The formula giving average value is $F_{a v g} (C) = \frac{{鈭�}_{C} F 路 r^{'} (t) d t}{{鈭�}_{C} ||r^{'} (t)|| d t}$

Step by step solution

01

Given Information

Average value for integrable function $f$ on interval $[a, b]$ is defined as:

role="math" localid="1653588435379" $f_{a v g} = \frac{1}{b - a} {鈭�}_{a}^{b} f (x) d x$

We need to generalize the expression to formula for finding the average value of $F (x, y, z)$ along a smooth curve $C$ parametrized by $r (t)$ for $a 鈮� t 鈮� b$ .

02

Simplifying the average value

Since $f_{avg} = \frac{1}{b - a} {鈭�}_{a}^{b} f (x) d x$

$= \frac{{鈭�}_{a}^{b} f (x) d x}{b - a}$

$= \frac{{鈭�}_{a}^{b} f (x) d x}{{鈭�}_{a}^{b} d x}$

03

Deduction of formula

The integral along smooth curve is

${鈭�}_{C} F (x, y, z) 路 d r$

The average value is

$F_{a v g} (C) = \frac{{鈭�}_{C} F (x, y, z) 路 d r}{Arc Length of C for a 鈮� t 鈮� b}$

$= \frac{{鈭�}_{C} F (x, y, z) 路 d r}{{鈭�}_{C} d s}$

$= \frac{{鈭�}_{C} (F 路 \frac{r^{'} (t)}{||r^{'} (t)||}) ||r^{'} (t)|| d t}{{鈭�}_{C} ||r^{'} (t)|| d t}$

$= \frac{{鈭�}_{C} F 路 r^{'} (t) d t}{{鈭�}_{C} ||r^{'} (t)|| d t}$

Hence, the required formula is $F_{a v g} (C) = \frac{{鈭�}_{C} F 路 r^{'} (t) d t}{{鈭�}_{C} ||r^{'} (t)|| d t}$

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