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 $a \leq t \leq b$ .

Short Answer

Expert verified

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

Step by step solution

Given Information

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

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

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 $a \leq t \leq b$ .

Simplifying the average value

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

$= \frac{\int_{a}^{b} f (x) d x}{b - a}$

$= \frac{\int_{a}^{b} f (x) d x}{\int_{a}^{b} d x}$

Deduction of formula

The integral along smooth curve is

$\int_{C} F (x, y, z) \cdot d r$

The average value is

$F_{a v g} (C) = \frac{\int_{C} F (x, y, z) \cdot d r}{Arc Length of C for a \leq t \leq b}$

$= \frac{\int_{C} F (x, y, z) \cdot d r}{\int_{C} d s}$

$= \frac{\int_{C} (F \cdot \frac{r^{'} (t)}{||r^{'} (t)||}) ||r^{'} (t)|| d t}{\int_{C} ||r^{'} (t)|| d t}$

$= \frac{\int_{C} F \cdot r^{'} (t) d t}{\int_{C} ||r^{'} (t)|| d t}$

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

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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 $a \leq t \leq b$ .

Short Answer

Step by step solution

Given Information

Simplifying the average value

Deduction of formula

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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≤t≤b.

Short Answer

Step by step solution

Given Information

Simplifying the average value

Deduction of formula

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Statistics

Applied Mathematics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

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 $a \leq t \leq b$ .