Q. 20 Given a differentiable vector fu... [FREE SOLUTION]

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Chapter 11: Q. 20 (page 872)

Given a differentiable vector function $r (t)$ defined on $[a, b]$ , explain why the integralrole="math" localid="1649610238144" ${鈭�}_{a}^{b} ||r^{'} (t)|| d t$ would be a scalar, not a vector.

Short Answer

Expert verified

By using the fundamental theorem of calculus, the definite integral is $f (b) - f (a)$ , which is a scalar.

Step by step solution

Step 1. Given data

We have a differentiable vector function $r (t)$ defined on $[a, b]$ ,

Here we have to explain why the integral ${鈭�}_{a}^{b} ||r^{'} (t)|| d t$ would be a scalar, not a vector.

Step 2. Explanation

Let us consider a differentiable vector function $r (t)$ with two components. on $[a, b]$ .

That is, $r (t) = 鉄� x (t), y (t) 鉄�$

$\begin{aligned} r^{鈥�} (t) = 鉄� x^{鈥�} (t), y^{鈥�} (t) 鉄� \\ 鈭� r^{鈥�} (t) 鈭� = \sqrt{(x^{鈥�} (t)^{2}) + {(y^{鈥�} (t))}^{2}} \\ 鈭� 鈭� r^{鈥�} (t) 鈭� d t = 鈭� \sqrt{(x^{鈥�} (t)^{2}) + {(y^{鈥�} (t))}^{2}} d t \end{aligned}$

This means here, $鈭� ||r^{'} (t)|| d t$ represents a function of $t$ but not a vector now.

${{鈭�}_{a}^{b} 鈭� r^{鈥�} (t) 鈭� d t = {[鈭� \sqrt{{(x^{鈥�} (t))}^{2} + {(y^{鈥�} (t))}^{2}}]}_{a}^{b} = [f (t)]}_{a}^{b}$

Here, by using the fundamental theorem of calculus, the definite integral is $f (b) - f (a)$ , which is a scalar.

Therefore, ${鈭�}_{a}^{b} 鈭� r^{鈥�} (t) 鈭� d t$ is always a scalar.

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Given a differentiable vector function $r (t)$ defined on $[a, b]$ , explain why the integralrole="math" localid="1649610238144" ${鈭�}_{a}^{b} ||r^{'} (t)|| d t$ would be a scalar, not a vector.

Short Answer

Step by step solution

Step 1. Given data

Step 2. Explanation

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Given a differentiable vector function r(t)defined on [a,b], explain why the integralrole="math" localid="1649610238144" 鈭�abr'(t)dt would be a scalar, not a vector.

Short Answer

Step by step solution

Step 1. Given data

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Calculus

Pure Maths

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Given a differentiable vector function $r (t)$ defined on $[a, b]$ , explain why the integralrole="math" localid="1649610238144" ${鈭�}_{a}^{b} ||r^{'} (t)|| d t$ would be a scalar, not a vector.