Q. 39 Find the curvature of each of th... [FREE SOLUTION]

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Chapter 11: Q. 39 (page 890)

Find the curvature of each of the vector-valued functions defined in Exercises 39鈥�44.
$r (t) = (t 鈭� \sin t, 1 鈭� \cos t)$

Short Answer

Expert verified

The curvature of the given vector-valued function is $魏 = \frac{1}{2 \sqrt{2 (1 - \cos t)}} .$

Step by step solution

Step 1. Given Information.

The given vector-valued function is $r (t) = (t 鈭� \sin t, 1 鈭� \cos t) .$

Step 2. Find the curvature.

To find the curvature of the given vector-valued function, we will use the formula for the Curvature of a Space Curve $魏 = \frac{||T' (t)||}{||r' (t)||} .$

So,

role="math" localid="1649666397890" $r (t) = (t - \sin t, 1 - \cos t) r' (t) = (1 - \cos t, \sin t) ||r' (t)|| = \sqrt{{(1 - \cos t)}^{2} + \sin^{2} t} ||r' (t)|| = \sqrt{1 + \cos^{2} t - 2 \cos t + + \sin^{2} t} ||r' (t)|| = \sqrt{2 (1 - \cos t)} ||r' (t)|| = 2 \sin \frac{t}{2} . . . . . . . . . . (a) Now, T (t) = \frac{r' (t)}{||r' (t)||} T (t) = \frac{<1 - \cos t, \sin t>}{2 \sin \frac{t}{2}} T (t) = <\frac{1 - \cos t}{2 \sin \frac{t}{2}}, \frac{\sin t}{2 \sin \frac{t}{2}}> T (t) = <\sin \frac{t}{2}, \cos \frac{t}{2}>$

Step 3. Calculate.

By proceeding with the calculation further,

$T' (t) = <\frac{1}{2} \cos \frac{t}{2}, \frac{- 1}{2} \sin \frac{t}{2}> ||T' (t)|| = \sqrt{{(\frac{1}{2} \cos \frac{t}{2})}^{2} + {(\frac{- 1}{2} \sin \frac{t}{2})}^{2}} ||T' (t)|| = \sqrt{\frac{1}{4} (\cos^{2} \frac{t}{2} + \sin^{2} \frac{t}{2})} ||T' (t)|| = \frac{1}{2} . . . . . . . . . . . (b)$

Put (a) and (b) in the formula of Curvature of a space curve,

$魏 = \frac{\frac{1}{2}}{2 \sin \frac{t}{2}} 魏 = \frac{1}{2 \sqrt{2 (1 - \cos t)}}$

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91影视

Find the curvature of each of the vector-valued functions defined in Exercises 39鈥�44.
$r (t) = (t 鈭� \sin t, 1 鈭� \cos t)$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the curvature.

Step 3. Calculate.

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91影视

Find the curvature of each of the vector-valued functions defined in Exercises 39鈥�44.r(t)=t鈭�sint,1鈭�cost

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the curvature.

Step 3. Calculate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Statistics

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the curvature of each of the vector-valued functions defined in Exercises 39鈥�44.
$r (t) = (t 鈭� \sin t, 1 鈭� \cos t)$