Scalar Line Integrals: Let x : [a, b] → R^{n} be a C^{1} path. f : R^{n} → R, a scalar field. Δs_{k} = length of kth segment of path

= ∫_{t_{k−1}}^{t_{k}} ||x′(t)||dt = ||x′(t_{k}^{∗∗})||(t_{k} − t_{k−1}) = ||x′(t_{k}^{∗∗})||Δt_{k}

for some t_{k}^{∗∗} ∈ [t_{k−1}, t_{k}]

∫_{x} f ds ∼ Σ_{i=1}^{n}f(x(t_{k}^{∗}))Δs_{k} = Σ_{i=1}^{n}f(x(t_{k}^{∗}))||x′(t_{k}^{∗∗})||Δt_{k} = ∫_{a}^{b} f(x(t))||x′(t)||dt

Vector Line integrals: Let x : [a, b] → R^{n} be a C^{1} path. F : R^{n} → R^{n}, a vector field. x′(t_{k}^{∗}) ∼ \frac{Δx_{k}}{Δt_{k}} ∫_{x} F · ds ∼ Σ_{i=1}^{n}F (x(t_{k}^{∗})) · Δx_{k} = Σ_{i=1}^{n}F (x(t_{k}^{∗})) · x′(t_{k}^{∗})Δt_{k}

= ∫_{a}^{b} F (x(t)) · x′(t)dt Other Formulations of Vector Line integrals: The tangent vector to x at t is T (t) = \frac{x′(t)}{||x′(t)||} ∫_{x} F · ds = ∫_{a}^{b} F (x(t)) · x′(t)dt = ∫_{a}^{b} F (x(t)) · x′(t)\frac{||x′(t)||}{||x′(t)||}dt = ∫_{a}^{b} F (x(t)) · T (t)||x′(t)||dt = ∫_{x} F (x(t)) · T (t)ds Note ∫_{a}^{b} F (x(t)) · T (t)ds is a scalar line integral of the scalar field F · T : R^{n} → R over the path x. Note: F ·T is the tangential component of F along the path x. Another notation (differential form): For simplicity, we will work in R^{2}, but the following generalizes to any dimension. Let x(t) = (x(t), y(t)). Let F (x, y) = (M(x, y), N(x, y)) x = x(t), y = y(t). x′(t) = \frac{dx}{dt}, y′(t) = \frac{dy}{dt}

∫_{x} F · ds = ∫_{a}^{b} F (x(t)) · x′(t)dt
= ∫_{a}^{b}(M(x, y), N(x, y)) · (x′(t), y′(t))dt
= ∫_{a}^{b} M(x, y)x′(t)dt + N(x, y)y′(t)dt
= ∫_{x} M(x, y)dy + N(x, y)dy

Definitions: A curve is the image of piecewise C^{1} path x : [a, b] → R^{n}. A curve is simple if it has no self-intersections; that is, x is 1:1 on the open interval (a, b) A path is closed if x(a) = x(b) A curve is closed if x(a) = x(b) ∫_{x} F · ds is called the circulation of f along x if x is a closed path.

A parametrization of a curve C is a path whose image is C. Normally we will require a parametrization of a curve to be 1:1 where possible.

A piecewise C^{1} path y : [c, d] → R^{n} is a reparametrization of a piecewise C^{1} path x : [a, b] → R^{n} if there exists a bijective C^{1} function u : [c, d] → [a, b] where the inverse of u is also C^{1} and y = x ∘ u (i.e., y(t) = x(u(t))).

Note that either

1.) u(a) = c and u(b) = d. In this case, we say that y (and u are orientation-preserving.

or

2.) u(a) = d and u(b) = c In this case, we say that y (and u are orientation-reversing.

Given piecewise C^{1} path, x : [a, b] → R^{n}, the opposite path is x_{opp} : [a, b] → R^{n} x_{opp} = x(a + b − t)

That is x_{opp} is a reparametrization of x where u[a, b] → [a, b], u(t) = a + b − t.

Note x_{opp} is an orientation-reversing reparametrization of x.