Line Integral

Topics: Calculus - Integral

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Naming

In all my notes, path integral refers to the integral of a scalar function over a curve, while line integral refers to the integral of a vector field over a curve.

Some authors call both line integrals (of a scalar field and of a vector field, respectively).

(definition)

Let $F$ be a vector field on ${\mathbb{R}}^3$ that is continuous on the curve $c:[a,b]\to {\mathbb{R}}^3$.

We define the line integral of $F$ along $c$ as:

$${\int }_{c}F\cdot ds:={\int }_a^b(F\circ c)(t)\cdot c^{\prime }(t)dt$$