Vector calculus is multivariate [real analysis]
? in (usually) 2 and 3 dimensions.
The field consists of a suite of formulas and problem solving techniques very useful for
engineering and
Newtonian physics.
Most of the analytic results are more easily understood using the machinery of
differential geometry, for which vector calculus forms a subset.
Vectors live in a space called a "vector space" over a given field? that overloads two operations (vector + vector and scalar * vector) to follow eight rules.
Given vectors u, v, and w, and scalars a and b:
- 0 + v = v
- v + w = w + v
- (u + v) + w = u + (v + w)
- 0 * v = 0
- 1 * v = v
- (a + b) * v = a * v + b * v
- a * (v + w) = a * v + a * w
- (help!)
Examples of vector spaces:
- F^n, over F
- R^n, over R (the real numbers)
- The finite field GF(p^n), over GF(p)
- C (complex numbers), over R
- Given a field F and a vector space V over F, the set of functions F -> V, over F
- R, over Q (the rational numbers).
Conjecture: If A is a vector space over B, and B is a vector space over C, A is a vector space over C.
(Proof? Disproof?)
Sources
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