Covariant, Absolute, and Contravariant Derivatives

Abstract

The curve represented by a function ϕ(x ⁱ) in a closed interval is continuous if this function is continuous in this interval. If the curve is parameterized, i.e., ϕ[x ⁱ(t)] being $t\in \left[a,b\right]$, then it will be continuous if x ⁱ(t) are continuous functions in this interval, and it will be smooth if it has continuous and non-null derivatives for a value of $t\in \left[a,b\right]$. The smooth curves do not intersect, i.e., the conditions ${x}^i(a)={x}^i(b)$ will only be satisfied if $a=b$. This condition defines a curve that can be divided into differential elements, forming curve arcs. For the case in which the initial and final points coincide, expressed by condition $a=b$, the curve is closed. The various differential elements obtained on the curve allow calculating its line integral.