# Line Integral
**The [[integral]] of a multivariable function along a curve as the domain of integration.**
![[LineIntegral.svg|500]]
*A line integral of a scalar field can be represented as the total signed area between the surface of a multivariable function and a curve.*
A *line integral* is the integral of a multivariable function, whether a scalar or a vector field, as evaluated *along a curve*. It is a generalisation of definite integrals to both multivariable functions and arbitrary curves.
A line integral is the *sum of all field values* along the curve, either multiplied by a differential arc length along the curve or the [[dot product]] of the vector field with a differential vector along the curve.
It is common to *parameterise* the curve in terms of a single variable, which converts the domain of integration into a closed interval on the real line.
A line integral of a single variable function over a straight line in the same plane as the graph of the function is the same as a definite integral, although with different notation.
## Definition
### Scalar field
> [!NOTE] Line integral of a scalar field
> For some scalar field $f$, the *line integral* along a piecewise smooth curve $C$ is defined as
> $\int_C f\;ds=\int _{a}^{b}f(\mathbf{r}(t))|\mathbf{r}'(t)|\;dt$
> where $\mathbf{r}:[a,b]\rightarrow C$ is a *parameterisation* of the curve such that $\mathbf{r}(a)$ and $\mathbf{r}(b)$ are the endpoints of $C$ and $a<b$.
The scalar field $f$ is the *integrand*, the curve $C$ is the *domain of integration*, and $ds$ represents a *differential arc length* along the curve.
The value of the line integral of a scalar field *does not depend* on the parameterisation used.
Intuitively, the line integral of a scalar field can be seen as analogous to a [[Riemann integral]], where the integrand function consists of the values of the scalar field above the curve and the interval of integration is the curve.
### Vector field
> [!NOTE] Line integral of a vector field
> For some vector field $\mathbf{F}$, the *line integral* along a piecewise smooth curve $C$ is defined as
> $\int_C \mathbf{F}(\mathbf{r})\cdot d\mathbf{r}=\int _{a}^{b}\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\;dt$
> where $\mathbf{r}:[a, b]\rightarrow C$ is a *parameterisation* of the curve such that $\mathbf{r}(a)$ and $\mathbf{r}(b)$ are the endpoints of $C$.
The absolute value of the line integral of a vector field *does not depend* on the parameterisation used, although a *reversal in orientation* will change its sign.
Intuitively, the line integral of a vector field is the sum of the tangential components of all vectors along the curve multiplied by the length of each differential arc.
#### Path independence
> [!NOTE] Path independence
> If a vector field $\mathbf{F}$ is *conservative*, that is, it is the [[gradient]] of a scalar field $G$, then any line integral on it equals the *difference between the values* of $G$ at the *endpoints* of the curve of integration $C$.
> $\int_C \mathbf{F}(\mathbf{r})\cdot d\mathbf{r}=G(\mathbf{r}(b)) - G(\mathbf{r}(a))$