Hello all. I am a 17-year-old high school student in the United States. I have taken BC Calculus and will be taking Multi-variable Calculus next year (also known as Calc III in the states), but I already know a lot of it for various reasons. I also have mild experience in topology and abstract algebra. This is to give you an idea of where I am regarding my math skills. I have little difficulty understanding new math concepts EXCEPT when there are undefined characters and strange notation floating around. I understand the basic concept of a tensor, and I find it fascinating. To me, what constitutes cool math is taking something you already get on one level and generalizing it to see what new cool stuff can come from it, so the idea of generalizing the notation of a scalar, vector and matrix to a quantity with n indices seems really cool to me. I just have a few questions slash statements: 1. My understanding of the difference between the tensor product and the dot product is that the dot product is matrix multiplication and the tensor product is laying out each component of two tensors and multiplying them together as though they were terms of seperate polynomials. Is this correct? 2. It seems to me like a particle in motion in Euclidian (or any) space could be modeled with a second-order tensor: one index would tell you which vector you are interested in (position, velocity, etc.) and the other index would tell you which spatial component of that vector you are interested in. Yet I have never encountered a particle being modeled in this way. Is this because the math involved with tensor analysis would never be useful to model a particle in this way, or am I being stupid? 3. Could someone explain in words the concepts of covariant and contravariant indices? I have not taken a course in linear algebra, and I understand that some (or all) of it relates to linear algebra, so "shut up about tensors until you've taken a course in linear algebra" is a completely acceptable answer to this question. But if there is a way to explain it in words to someone that hasn't taken a course in linear algebra, I would appreciate hearing it. I have seen the formal definition that involves the word "transformation," a product of partial derivatives, some undefined x's, bars, and indices, all of which mean nothing to me, so repeating the formal definition found anywhere online would not be helpful. Thanks.