Give the function we’re working with a name (r), and see some simple examples of using that name in places where you want the function:

r = { Sqrt[3]*Cos[t], Cos[t], 2 * Sin[t] }

r

ParametricPlot3D[ r, {t,0, 2*Pi}]

Name the various quantities needed to apply a curvature formula...

rPrime = D[ r, t ]

Mathematica has a “Norm” function that computes the magnitude of a vector.

rPrimeMag = Norm[ rPrime ]

Computer tools don’t always do things the way you want them to. So pay attention to what the tool says vs what you expect, be sure to reconcile differences, and feel free to give the tool a better answer to something if you have one (like here):

rPrimeMag = 2

Keep calculating and naming other things needed for curvature.

unitT = rPrime / rPrimeMag

unitTPrime = D[ unitT, t ]

unitTPrimeMag = Norm[ unitTPrime ]

unitTPrimeMag = 1

Use the |T’|/|r’| formula to finally calculate curvature.

kappa = unitTPrimeMag / rPrimeMag