Line Integral - Path/Curve/Curvilinear Integral

Line Integral of a Scalar-Valued Function

For some scalar-valued function 𝑓: ℝ^𝑛 → ℝ, the line integral along a piecewise smoothcurve 𝐶⊂ℝ^𝑛 is defined as:

• ${\int }_{C}f(\mathbf{r})ds={\int }_a^{b}f(\mathbf{r}(t))|{\mathbf{r}}^{\prime }(t)|dt$

where:

• 𝐫: [𝑎,𝑏] → 𝐶 is an arbitrary bijectiveparametrization of the curve 𝐶 such that 𝐫(𝑎) and 𝐫(𝑏) give the endpoints of 𝐶 and 𝑎<𝑏
• 𝑑𝑠 may be interpreted as an elementary arc length of curve 𝐶

2D Example

The line integral over a scalar-valued function 𝑓 can be thought of as the area under the curve 𝐶 along a surface 𝑧 = 𝑓(𝑥,𝑦), described by the field

Line Integral of a Vector-Valued Function

For some vector-valued function 𝑓: ℝ^𝑛 → ℝ^𝑛, the line integral along a piecewise smoothcurve 𝐶⊂ℝ^𝑛 in direction 𝑟 is defined as:

• ${\int }_{C}f(\mathbf{r})d\mathbf{r}={\int }_a^{b}f(\mathbf{r}(t))\cdot |{\mathbf{r}}^{\prime }(t)|dt$

where:

• 𝑟 represents the dot product

Resources

• https://www.youtube.com/watch?v=Tz14rC0XvHI