Chapter 11: Q. 43 (page 890)

Find the curvature of each of the vector-valued functions defined in Exercises 39–44.
$r (t) = (t \sin t, t \cos t, α t), where α is a constant$

Short Answer

Expert verified

The curvature of the given vector-valued function is $κ = \sqrt{\frac{t^{4} + (4 + α^{2}) t^{2} + 4 + 4 α^{2}}{{(t^{2} + 1 + α^{2})}^{3}}} .$

Step by step solution

Step 1. Given Information.

The given vector-valued function is $r (t) = (t \sin t, t \cos t, α t), where α is a constant .$

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 $κ = \frac{||r' (t) \times r'' (t)||}{{||r' (t)||}^{3}} .$

So,

$r (t) = <t \sin t, t \cos t, α t> r' (t) = <t \cos t + \sin t, - t \sin t + \cos t, α> r'' (t) = <- t \sin t + 2 \cos t, - t \cos t - 2 \sin t, 0> Now, r' (t) \times r'' (t) = |\begin{matrix} i & j & k \\ t \cos t + \sin t & - t sint & α \\ - t \sin t + 2 \cos t & - t \cos t - 2 \sin t & 0 \end{matrix}| = i [α (t \cos t + 2 \sin t)] - j [α (t \sin t - 2 \cos t)] + k (- t^{2} - 2) = <α (t \cos t + 2 \sin t), α (t \sin t - 2 \cos t), - t^{2} - 2> Let' s find ||r' (t) \times r'' (t)|| = \sqrt{{(α (t cost + 2 sint))}^{2} + {(α (- t sint + 2 cost))}^{2} + {(- t^{2} - 2)}^{2}} = \sqrt{t^{4} + (4 + α^{2}) t^{2} + 4 + 4 α^{2}} . . . . . . . . . (a) And ||r' (t)|| = \sqrt{{(t \cos t + \sin t)}^{2} + {(- t sint + \cos t)}^{2} + {(α)}^{2}} = \sqrt{t^{2} + 1 + α^{2}} . . . . . . . . (b)$

Step 3. Calculate.

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

$κ = \frac{\sqrt{t^{4} + (4 + α^{2}) t^{2} + 4 + 4 α^{2}}}{{(\sqrt{t^{2} + 1 + α^{2}})}^{3}} κ = \sqrt{\frac{t^{4} + (4 + α^{2}) t^{2} + 4 + 4 α^{2}}{{(t^{2} + 1 + α^{2})}^{3}}}$

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Find the curvature of each of the vector-valued functions defined in Exercises 39–44.
$r (t) = (t \sin t, t \cos t, α t), where α is a constant$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find the curvature.

Step 3. Calculate.

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Find the curvature of each of the vector-valued functions defined in Exercises 39–44.r(t)=tsint,tcost,αt,whereαisaconstant

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

Discrete Mathematics

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Mechanics Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Find the curvature of each of the vector-valued functions defined in Exercises 39–44.
$r (t) = (t \sin t, t \cos t, α t), where α is a constant$