Geometrical vectors

What Hacker News says

1. bmitcMar 2022
   Anyone interested in a visual exploration should checkout Geometrical Vectors by Gabriel Weinreich.

   https://www.maa.org/press/maa-reviews/geometrical-vectors

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2. martincmartinJul 2020
   The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points move closer together, then the distance vector gets smaller. However, the gradient vector gets bigger. The gradient is basically a density, it's units are units-of-whatever-your-taking-the-gradient-of / meters.

   And, or course, the vector you get from a cross product only has 3 degrees of freedom because we live in a 3D world. The cross product of two 2D vectors is a scalar, the cross product of two 4D vectors needs 6 numbers to describe it. Even in 3D, if you reflect the original vectors in a mirror, the cross product now points the opposite direction, i.e. it depends on the handednes of the coordinate system.

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3. bmitcMay 2020
   I listed references elsewhere.

   https://news.ycombinator.com/item?id=23270163

   Edwards' first three chapters give a wonderfullly intuitive exposition of forms and their application to integration.

   Tu's book is a rigourous study of smooth manifolds and differential forms. His exercises are approachable, and his book is the most expedient to the full theory of differential forms.

   As a quirky intuition pump, I recommend Geometrical Vectors by Gabriel Weinreich. The Fortney book mentioned in another comment is a nice, visual book, and there are other references in the replies to the comment I linked.

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