Smooth curve

A smooth curve is a curve parameterized by a vector-valued function $\mathbf{r}(t)$ where ${\mathbf{r}}^{\prime }(t)$ exists, is continuous, and is nonzero everywhere on the parameter interval.

The three conditions matter:

• ${\mathbf{r}}^{\prime }$ exists: no kinks or jumps in position.
• ${\mathbf{r}}^{\prime }$ continuous: the tangent direction varies smoothly.
• ${\mathbf{r}}^{\prime }\ne 0$: no stalls. A zero derivative would mean the parameter point momentarily stops, after which the tangent direction could change abruptly even though the path itself is geometrically smooth.

The condition ${\mathbf{r}}^{\prime }\ne 0$ is sometimes phrased as “regular” rather than “smooth.” Different textbooks differ slightly in convention. Vector Calculus and Complex Analysis calls a curve smooth when all three conditions hold.

Why the nonzero requirement

Consider $\mathbf{r}(t)=⟨t^3,t^2⟩$. The image is the cusp-shaped curve $y=x^{2/3}$, which has a corner at the origin even though $\mathbf{r}$ is infinitely differentiable. The issue: ${\mathbf{r}}^{\prime }(0)=⟨0,0⟩$, so the parameterization stalls at the origin, and the curve can change tangent direction without the parameterization “noticing.”

Requiring ${\mathbf{r}}^{\prime }\ne 0$ rules out such corners and ensures a well-defined unit tangent vector everywhere.

Piecewise smooth

A piecewise smooth curve is built from finitely many smooth pieces joined end-to-end at corners. Examples:

• Boundary of a triangle: three smooth line segments joined at three corners.
• An L-shaped path from $(0,0)$ to $(2,0)$ to $(2,4)$.
• The boundary of a square or rectangle.

Piecewise smoothness is the standard assumption for line integrals in vector calculus. Theorems like Green’s theorem and Stokes’ theorem all assume the boundary curve is piecewise smooth.

To integrate over a piecewise smooth curve, break it into smooth pieces and add the integrals of each piece.

In the bigger picture

Smoothness is the regularity needed to make calculus work cleanly on curves:

• Differentiation gives a well-defined tangent direction.
• Arc length is well-defined.
• Line integrals over the curve are well-defined and don’t depend on the choice of parameterization within the same orientation.

For non-smooth curves (fractals, the von Koch snowflake, things with corners or stalls), more general machinery is needed — Lebesgue integration, rectifiability, measure theory. Vector Calculus and Complex Analysis stays in the smooth / piecewise smooth setting.