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CAE_Jones comments on Self-Study Questions Thread - Less Wrong Discussion
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Somehow, the way my education jumped about, I got through high school, acing advanced topics in physics, astrophysics, and scoring high enough on the advanced placement calculus exam to skip some college courses, all without learning the cross or dot products.
I didn't do so well in college physics.
(I tried researching these online, but everything I found was either impenetrably technical or used an inaccessible format for mathematic notation. I finally got a hold of javascript functions, but I still wouldn't know when to use them.)
Div, Grad, Curl is a great introduction to vector calculus.
If you're still interested in learning, I found that physics 1 on MIT OCW did a pretty good job of explaining it.
To just learn the operations, this was a good reference. Dot product and cross product is explained about halfway through. here is the accompanying video where Walter Lewin (a charismatic and energetic physics prof) walks through actually doing it.