📂Calculus

Line Integrals of Vector Fields

Definition¹

Let a vector field $\mathbf{F} : \mathbb{R}^{3} \to \mathbb{R}^{3}$ and a curve $C$ in 3-dimensional space be given as $\mathbf{r}(t)$. Let $\mathbf{T}$ be called the tangent field of the vector field. Then, the $\mathbf{F}$ line integral along the curve $C$ is defined as follows.

$$\int_{C} \mathbf{F} \cdot d \mathbf{r} = \int_{a}^{b} \mathbf{F}\left( \mathbf{r}(t) \right) \cdot \mathbf{r}^{\prime}(t) dt = \int_{C} \mathbf{F} \cdot \mathbf{T} ds$$

Explanation

The buildup to defining the line integral of a vector field is no different from that of defining the length of a curve or the line integral of a scalar field, so refer to those.

Physical Meaning

If the vector field $\mathbf{F}$ represents a force and the curve $C$ represents the path along which an object has moved, then the line integral of the vector field is work itself.

$$W = \int_{C} \mathbf{F} \cdot \mathbf{T} ds$$