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4

Natas' answer is almost correct, and I gave it an up vote. However, technically what Grad computes is the raised covariant derivative $\nabla^b T^{cd\ldots} = g^{ba}\partial_aT^{cd\ldots} + \Gamma^{bc}_aT^{ad\ldots} + \ldots$. The beauty of orthonormal bases, and the reason they're the only ones exposed in System` functionality, is that components are ...

1

It is not clear if $k_z$ depends on z or not. In both cases, you can try writing in the following way: f[r_, \[Theta]_, z_] := BesselJ[0, r Sqrt[\[Omega]^2/c^2 - k[z]^2]] S[k[z]] Exp[I k[z] z]; Grad[f[r, \[Theta], z], {r, \[Theta], z}, "Cylindrical"] with $k[z]$ taking into account for the dependance. The output is the gradient in cylindrical ...

0

r[t_] := {t, 0.1 t^2, 0.1 t^3} T[t_] := Normalize[r'[t]] n[t_] := Normalize[T'[t]] T[t] ({1/Sqrt[1 + 0.04 RealAbs[t]^2 + 0.09 RealAbs[t]^4], (0.2 t)/Sqrt[ 1 + 0.04 RealAbs[t]^2 + 0.09 RealAbs[t]^4], (0.3 t^2)/Sqrt[ 1 + 0.04 RealAbs[t]^2 + 0.09 RealAbs[t]^4]}) n[t] ({-((0.08 Abs[t] Derivative[Abs][t] + 0.36 Abs[t]^3 Derivative[Abs][t])/(2 (1 + 0.04 ...

1

According to Alfred Gray's Differential Geometry book,it is recommend to use the following way to calculate the torsion . r[t_] := {t, 0.1 t^2, 0.1 t^3}; T[t_] := Normalize[r'[t]]; B[t_] := Normalize[Cross[r'[t], r''[t]]]; n[t_] := Cross[B[t], T[t]]; Torsion[t_] := Det[{r'[t], r''[t], r'''[t]}]/Norm[Cross[r'[t], r''[t]]]^2 N[Torsion[3.16]] (* 0.0300467 *)

6

Consider: T[t] {1/Sqrt[1 + 0.04 Abs[t]^2 + 0.09 Abs[t]^4], (0.2 t)/Sqrt[ 1 + 0.04 Abs[t]^2 + 0.09 Abs[t]^4], (0.3 t^2)/Sqrt[ 1 + 0.04 Abs[t]^2 + 0.09 Abs[t]^4]} As you can see, it contains the function "Abs". In complex numbers, Abs is nowhere differentiable. And MMA assumes, without being told otherwise, that all numbers are complex. Because of ...

0

Since your function V[t] uses x[t],y[t],z[t] (okay x is not used but anyways) I would define them first using this code: x[t_] = Sin[t^2]; y[t_] = t^2 - Cos[t]; z[t_] = Sinh[t] - Cos[t]; Now you can define V[t] itself V[t_] = {2, y[t]*z[t]^2, 3*y[t] + z[t]}; We can now use ParametricPlot3D : ParametricPlot3D[ V[t], {t, 0, 2 Pi}, PlotStyle -> ...

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