- #1

- 155

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Obviously the two choices for E_1 and E_2 are the unit normal and unit tangent vectors to the curve.

Using Mathematica...

Or by hand...alpha[t_] := {Cos[t], 2 Sin[t]};

alphaprime[t_] := {-Sin[t], 2 Cos[t]};

alphaprimeprime[t_] := {-Cos[t], -2 Sin[t]};

unittangentvector[t_] := alphaprime[t] / Norm[alphaprime[t]];

unitnormalvector[t_] := alphaprimeprime[t] / Norm[alphaprimeprime[t]];

[itex]\alpha(t) = {Cos(t), 2Sin(t)}[/itex]

[itex]\alpha'(t) = {-Sin(t), 2Cos(t)}[/itex]

[itex]\alpha''(t) = {-Cos(t), -2Sin(t)}[/itex]

However, graphically, the unit tangent and unit normal vectors are far from perpendicular on this curve!

Here is my mathematica code

VFieldOnCurve2D[dominterval_, CurveEq_, FrameField_, CodomainBox_,

Size_] :=

Module[{a2, b2, Content, IS, DomainPieces, DomainPiece1,

DomainPiece2, CodomainCenter, CodomainWidth, len, EE1, EE2,

ImagePieces, ImagePiece0, ImagePiece1, ImagePiece2},

{a2, b2} = dominterval;

IS = 300;

Content = Mapping12Content[dominterval, CurveEq];

DomainPiece1 = Content[[1]];

DomainPiece2[t_] := Points2D[{{0, t}}, .3];

DomainPieces[t_] :=

Show[DomainPiece1, DomainPiece2[t], ImageSize -> IS/4];

{CodomainCenter, CodomainWidth} = CodomainBox;

len = Length[FrameField];

If[len == 2, EE1 = FrameField[[1]];

EE2 = FrameField[[2]], {EE1} = FrameField];

ImagePiece0 = EmptySpace2DXCenter[CodomainCenter, CodomainWidth];

ImagePiece1 = Content[[2]];

ImagePiece2[t_] :=

If[len == 2, {Vec[CurveEq[t], EE1[t]], Vec[CurveEq[t], EE2[t]]},

Vec[CurveEq[t], EE1[t]] ];

ImagePieces[t_] :=

Show[ImagePiece0, ImagePiece1, ImagePiece2[t], ImageSize -> Size];

t0 = (a2 + b2)/2;

Manipulate[

Row[{DomainPieces[t], ImagePieces[t]}], {{t, t0, "t"}, a2, b2},

SaveDefinitions -> True]

]

alpha[t_] := {Cos[t], 2 Sin[t]};

alphaprime[t_] := {-Sin[t], 2 Cos[t]};

alphaprimeprime[t_] := {-Cos[t], -2 Sin[t]};

unittangentvector[t_] := alphaprime[t] / Norm[alphaprime[t]];

unitnormalvector[t_] := alphaprimeprime[t] / Norm[alphaprimeprime[t]];

E1[t_] = unitnormalvector[t];

E2[t_] = unittangentvector[t];

DomainInterval = {0, 2 \[Pi]};

initvalue = 0;

CodomainBox = {Origin2D, 2};

Size = 400;

VFieldOnCurve2D[DomainInterval, alpha, {E1, E2}, CodomainBox, Size]