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Review of Vector Algebra and Vector Calculus

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Preface: I'm writing this page because I have an exam over the material soon and I think it would be a good idea for me to review it like this.

Vector Algebra

What is a vector?

Physical quantities are primarily represented by two distinct classes of objects: scalars and vectors. Scalars only deal with magnitude. They can be real, complex, and anything inbetween. The number 5 is a scalar, e is a scalar, and 1 - 10j is a scalar. Vectors however, not only have a magnitude but also a direction in space. I will try my best to write vectors in Bold or with an arrow over them (in the case I end up using LaTeX). Scalars should not be boldened.