Summary

DescriptionCovariantcomponents.gif	English: Covariant and contravariant components of a vector when the basis is not orthogonal.
Date	8 December 2018
Source	https://twitter.com/j_bertolotti/status/1071417492692709376   This diagram was created with Mathematica. Category:PNG created with Mathematica#Covariantcomponents.gif
Author	Jacopo Bertolotti
Permission (Reusing this file)	https://twitter.com/j_bertolotti/status/1030470604418428929

Mathematica 11.0 code

line[p1_, p2_] := 
  Block[{m, q}, {m, q} /. 
    Solve[{p1[[2]] == m p1[[1]] + q, p2[[2]] == m p2[[1]] + q}, {m, 
       q}][[1]]];
intersection[l1_, l2_] := 
  Block[{x, y}, {x, y} /. 
    Solve[{y == l1[[1]] x + l1[[2]], y == l2[[1]] x + l2[[2]]}, {x, 
       y}][[1]] ];
parallel[l1_, p1_] := {l1[[1]], p1[[2]] - l1[[1]] p1[[1]]};
normal[l1_, p1_] := {-1/l1[[1]], p1[[2]] + 1/l1[[1]] p1[[1]]};
o = {0, 0};
pe1 = {1, 10^-5}/Norm[{1, 0.1}];
v = {2.1, 1};
plots = Table[
   pe2 = {10^-5 + j, 1}/Norm[{10^-5 + j, 1}];
   GraphicsRow[{
     Graphics[{
       Thick, Dashed, Gray, Line[{o, 3 pe1}], Line[{o, 3*pe2}], 
       Line[{v, 
         intersection[normal[line[o, pe1], v], line[o, pe1]]}], 
       Line[{v, 
         intersection[normal[line[o, pe2], v], line[o, pe2] ]}]
       ,
       Dashing[None], Black, Arrow[{o, pe1}], Arrow[{o, pe2}], Blue, 
       Arrow[{o, v}]
       ,
       Black, Disk[o, 0.05], 
       Disk[intersection[normal[line[o, pe2], v], line[o, pe2] ], 
        0.05], Disk[
        intersection[normal[line[o, pe1], v], line[o, pe1] ], 0.05],
       Text[Style["O", Bold, FontSize -> 14], o - {0.2, 0.2}], 
       Text[Style[
         "\!\(\*SubscriptBox[OverscriptBox[\(e\), \(^\)], \(x\)]\)", 
         Bold, FontSize -> 14], pe1 - {0, 0.15}], 
       Text[Style[
         "\!\(\*SubscriptBox[OverscriptBox[\(e\), \(^\)], \(y\)]\)", 
         Bold, FontSize -> 14], pe2 + {0, 0.15}], 
       Text[Style["\!\(\*SubscriptBox[\(v\), \(x\)]\)", Bold, 
         FontSize -> 14], 
        intersection[normal[line[o, pe1], v], line[o, pe1] ] + {0.2, 
          0.2}], Text[
        Style["\!\(\*SubscriptBox[\(v\), \(y\)]\)", Bold, 
         FontSize -> 14], 
        intersection[normal[line[o, pe2], v], line[o, pe2] ] + {0.1, 
          0.2}],
       Blue, 
       Text[Style["\!\(\*OverscriptBox[\(v\), \(\[Rule]\)]\)", Bold, 
         FontSize -> 14], v + {0.1, 0.1}]
       }, PlotRange -> {{-0.5, 2.5}, {-0.5, 2}}, 
      PlotLabel -> "Covariant components", 
      LabelStyle -> {Black, Bold}, ImageSize -> Medium]
     ,
     Graphics[{
       Thick, Dashed, Gray, Line[{o, 3 pe1}], Line[{o, 3*pe2}], 
       Line[{v, 
         intersection[parallel[line[o, pe1], v], line[o, pe2]]}], 
       Line[{v, 
         intersection[parallel[line[o, pe2], v], line[o, pe1] ]}]
       ,
       Dashing[None], Black, Arrow[{o, pe1}], Arrow[{o, pe2}], Blue, 
       Arrow[{o, v}]
       ,
       Black, Disk[o, 0.05], 
       Disk[intersection[parallel[line[o, pe1], v], line[o, pe2] ], 
        0.05], Disk[
        intersection[parallel[line[o, pe2], v], line[o, pe1] ], 
        0.05],
       Text[Style["O", Bold, FontSize -> 14], o - {0.2, 0.2}], 
       Text[Style[
         "\!\(\*SubscriptBox[OverscriptBox[\(e\), \(^\)], \(x\)]\)", 
         Bold, FontSize -> 14], pe1 - {0, 0.15}], 
       Text[Style[
         "\!\(\*SubscriptBox[OverscriptBox[\(e\), \(^\)], \(y\)]\)", 
         Bold, FontSize -> 14], pe2 + {0, 0.15}], 
       Text[Style["\!\(\*SubscriptBox[\(v\), \(x\)]\)", Bold, 
         FontSize -> 14], 
        intersection[parallel[line[o, pe2], v], line[o, pe1] ] + {0.2,
           0.2}], Text[
        Style["\!\(\*SubscriptBox[\(v\), \(y\)]\)", Bold, 
         FontSize -> 14], 
        intersection[parallel[line[o, pe1], v], line[o, pe2] ] + {0.1,
           0.2}],
       Blue, 
       Text[Style["\!\(\*OverscriptBox[\(v\), \(\[Rule]\)]\)", Bold, 
         FontSize -> 14], v + {0.1, 0.1}]
       }, PlotRange -> {{-0.5, 2.5}, {-0.5, 2}}, 
      PlotLabel -> "Contravariant components", 
      LabelStyle -> {Black, Bold}]
     }]
   , {j, 0, 1.8, 0.05}];
ListAnimate[Join[plots, Reverse@plots]]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.

The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse

Category:CC-Zero#Covariantcomponents.gif

This file, which was originally posted to https://twitter.com/j_bertolotti/status/1030470604418428929, was reviewed on 10 December 2018 by reviewer Ronhjones, who confirmed that it was available there under the stated license on that date.