\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{\mathrm{sec} } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{\mathrm{arccot} } \) \( \newcommand{\arcsec}{\mathrm{arcsec} } \) \( \newcommand{\arccsc}{\mathrm{arccsc} } \) \( \newcommand{\sech}{\mathrm{sech} } \) \( \newcommand{\csch}{\mathrm{csch} } \) \( \newcommand{\arcsinh}{\mathrm{arcsinh} } \) \( \newcommand{\arccosh}{\mathrm{arccosh} } \) \( \newcommand{\arctanh}{\mathrm{arctanh} } \) \( \newcommand{\arccoth}{\mathrm{arccoth} } \) \( \newcommand{\arcsech}{\mathrm{arcsech} } \) \( \newcommand{\arccsch}{\mathrm{arccsch} } \)

17Calculus Vector Functions - Curvature

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For the following discussion, we will consider a parameterized curve defined by the vector function \(\displaystyle{ \vec{r}(t) = \langle x(t), y(t), z(t) \rangle }\) which is traversed once on the continuous interval \( a \leq t \leq b \).
Some books use the Greek letter \(\kappa\) (kappa) for curvature. We use a capital K.

The curvature of a smooth curve is a measure of how 'tight' or 'sharp' the curve is. If we have a smooth curve \(\vec{r}\) and we have a function s which is the arc length function, the curvature is defined to be
\(\displaystyle{ K(s) = \left\| \frac{d\vec{T}}{ds} \right\|}\).

This equation for the curvature is not particularly useful for calculations. So we have several other ways to write the equation of the curvature. But first notice, that the curvature is a scalar function, not a vector function. And since it is the norm of a vector, the curvature will always be positive.

Curvature Formula #1

For our first equation to use when calculating the curvature, we will use the chain rule to write \(\displaystyle{ \frac{d\vec{T}}{dt} = \frac{d\vec{T}}{ds} \cdot \frac{ds}{dt} }\). We can solve for \(d\vec{T}/ds\) to get

\(\displaystyle{ K = \left\| \frac{d\vec{T}}{ds} \right\| = \frac{\| d\vec{T}/dt \|}{ \| ds/dt \| } = \frac{1}{\| \vec{v} \|} \left\| \frac{d\vec{T}}{dt} \right\| }\)

Notice in the previous equation, we used \(ds/dt = \|\vec{v}\|\) to simplify the equation somewhat.
Now we can write the first curvature formula in a form that we can use for calculations.

\(\displaystyle{ K(t) = \frac{1}{\|\vec{v}\|} \left\| \frac{d\vec{T}}{dt} \right\| = \frac{\| \vec{T}'(t) \|}{\|\vec{r}'(t)\|} }\)

Curvature Formula #2

If we define a vector \(\vec{a} = d\vec{v}/dt\) as an acceleration vector, a second curvature formula is
\(\displaystyle{ K=\frac{\|\vec{v} \times \vec{a} \|}{\|\vec{v}\|^3} }\)

Before working some practice problems, here is a quick video clip for you that should help you understand curvature a bit better.

MIP4U - Determining Curvature of a Curve Defined by a Vector Valued Function [3mins-9secs]

video by MIP4U

Practice

Instructions - - Unless otherwise instructed, calculate the curvature of these vector functions. Give your answers in exact, completely factored form.

\( \vec{r}(t) = 3t\vhat{i} + 4\sin t\vhat{j} + \) \( 4\cos t\vhat{k} \)

Problem Statement

\( \vec{r}(t) = 3t\vhat{i} + 4\sin t\vhat{j} + \) \( 4\cos t\vhat{k} \)

Final Answer

\( K(t) = 4/25 \)

Problem Statement

\( \vec{r}(t) = 3t\vhat{i} + 4\sin t\vhat{j} + \) \( 4\cos t\vhat{k} \)

Solution

2073 video

video by Krista King Math

Final Answer

\( K(t) = 4/25 \)

close solution
\( \vec{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle \)

Problem Statement

\( \vec{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle \)

Final Answer

\( K(t) = 1/2 \)

Problem Statement

\( \vec{r}(t) = \langle 2\cos(t), 2\sin(t) \rangle \)

Solution

2074 video

video by MIP4U

Final Answer

\( K(t) = 1/2 \)

close solution
Find the curvature at \(t=1\) for \( \vec{r}(t) = \langle t, t^2/2, t^3/3 \rangle \).

Problem Statement

Find the curvature at \(t=1\) for \( \vec{r}(t) = \langle t, t^2/2, t^3/3 \rangle \).

Final Answer

\( K(1) = \sqrt{2}/3 \)

Problem Statement

Find the curvature at \(t=1\) for \( \vec{r}(t) = \langle t, t^2/2, t^3/3 \rangle \).

Solution

2075 video

video by MIP4U

Final Answer

\( K(1) = \sqrt{2}/3 \)

close solution
Find the curvature of \( \vec{r}(t) = \langle -5t, 2t^3, 3t^4 \rangle \) at \( t = 2 \).

Problem Statement

Find the curvature of \( \vec{r}(t) = \langle -5t, 2t^3, 3t^4 \rangle \) at \( t = 2 \).

Final Answer

\( K(t) = 24\sqrt{3229}/(9817)^{3/2} \)

Problem Statement

Find the curvature of \( \vec{r}(t) = \langle -5t, 2t^3, 3t^4 \rangle \) at \( t = 2 \).

Solution

2076 video

video by MIP4U

Final Answer

\( K(t) = 24\sqrt{3229}/(9817)^{3/2} \)

close solution
Find the curvature of \( \vec{r}(t) = \langle 2\cos(3t),2\sin(3t), \) \(4t \rangle \) at \( t = 0 \).

Problem Statement

Find the curvature of \( \vec{r}(t) = \langle 2\cos(3t),2\sin(3t), \) \(4t \rangle \) at \( t = 0 \).

Final Answer

9/26

Problem Statement

Find the curvature of \( \vec{r}(t) = \langle 2\cos(3t),2\sin(3t), \) \(4t \rangle \) at \( t = 0 \).

Solution

This problem is solved in each of these videos. The first video uses the cross product version of the curvature formula, the second video does not use the cross product.

2077 video

video by MIP4U

2077 video

video by MIP4U

Final Answer

9/26

close solution

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