# Find the curvature K of the curve at the point r(t)=e^tcos tcdot i+e^tsin tcdot j+e^tk,P(1,0,1)

Find the curvature K of the curve at the point $r\left(t\right)={e}^{t}\mathrm{cos}t\cdot i+{e}^{t}\mathrm{sin}t\cdot j+{e}^{t}k,P\left(1,0,1\right)$
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$r\left(t\right)={e}^{t}\mathrm{cos}t\cdot i+{e}^{t}\mathrm{sin}t\cdot j+{e}^{t}k$
Differentiate
${r}^{\prime }\left(t\right)={e}^{t}\left(\mathrm{cos}t-\mathrm{sin}t\right)i+{e}^{t}\left(\mathrm{sin}t+\mathrm{cos}t\right)j+{e}^{t}k$
Find Magnitude
$|{r}^{\prime }\left(t\right)|={e}^{t}\sqrt{\left(\mathrm{cos}t-\mathrm{sin}t{\right)}^{2}+\left(\mathrm{sin}t+\mathrm{cos}t{\right)}^{2}+1}$
$|{r}^{\prime }\left(t\right)|={e}^{t}\sqrt{\left({\mathrm{cos}}^{2}t-{\mathrm{sin}}^{2}t-2\mathrm{cos}t\mathrm{sin}t{\right)}^{2}+\left({\mathrm{sin}}^{2}t+{\mathrm{cos}}^{2}t+2\mathrm{cos}t\mathrm{sin}t{\right)}^{2}+1}$
$|{r}^{\prime }\left(t\right)|={e}^{t}\sqrt{3}$
Recall that:$T\left(t\right)=\frac{{r}^{\prime }\left(t\right)}{|{r}^{\prime }\left(t\right)|}$
Therefore$T\left(t\right)=\frac{\left({e}^{t}\left(\mathrm{cos}t-\mathrm{sin}t\right)i+{e}^{t}\left(\mathrm{sin}t+\mathrm{cos}t\right)j+{e}^{t}k\right)}{\left({e}^{t}\sqrt{3}\right)}$
$T\left(t\right)=\frac{1}{\sqrt{3}\left[\left(\mathrm{cos}t-\mathrm{sin}t\right)i+\left(\mathrm{sin}t+\mathrm{cos}t\right)j+k\right]}$
Differentiate${T}^{\prime }\left(t\right)=\frac{1}{\sqrt{3}\left[\left(-\mathrm{sin}t-\mathrm{cos}t\right)i+\left(\mathrm{cos}t-\mathrm{sin}t\right)j+0k\right]}$
Find Magnitude$|{T}^{\prime }\left(t\right)|=\frac{1}{\sqrt{3}\sqrt{\left(-\mathrm{sin}t-\mathrm{cos}t{\right)}^{2}+\left(\mathrm{cos}t-\mathrm{sin}t{\right)}^{2}+0\right)}}$
$|{T}^{\prime }\left(t\right)|=\frac{1}{\sqrt{3}\sqrt{\left({\mathrm{sin}}^{2}t+{\mathrm{cos}}^{2}t+2\mathrm{sin}t\mathrm{cos}t\right)+\left({\mathrm{cos}}^{2}t+{\mathrm{sin}}^{2}t-2\mathrm{cos}t\mathrm{sin}t\right)+0\right)}}$
$|{T}^{\prime }\left(t\right)|=\frac{\sqrt{2}}{\sqrt{3}}$
Recall that$k\left(t\right)=\frac{\frac{\sqrt{2}}{\sqrt{3}}}{\left({e}^{t}\sqrt{3}\right)}=\frac{\sqrt{2}}{\left(3{e}^{t}\right)}$
Result
$k\left(t\right)=\frac{\sqrt{2}}{\left(3{e}^{t}\right)}$