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Smooth Curve — Definition, Formula & Examples

A smooth curve is a curve that has a continuous derivative at every point, meaning it has no sharp corners, cusps, or breaks. Intuitively, you can draw it without lifting your pen and without any sudden changes in direction.

A parametric curve r(t)=(x(t), y(t))\mathbf{r}(t) = (x(t),\, y(t)) defined on an interval [a,b][a, b] is smooth if x′(t)x'(t) and y′(t)y'(t) are continuous on [a,b][a, b] and the derivative r′(t)≠0\mathbf{r}'(t) \neq \mathbf{0} for all t∈(a,b)t \in (a, b). For a function y=f(x)y = f(x), the curve is smooth on an interval if f′(x)f'(x) exists and is continuous throughout that interval.

How It Works

To determine whether a curve is smooth, check two things: first, that the derivative exists at every point in the domain, and second, that the derivative is continuous (no jumps). For parametric curves, you also need the derivative vector to never be the zero vector, because a vanishing derivative can indicate a cusp. A curve that is smooth on separate pieces but not everywhere is called piecewise smooth — each piece is smooth, but they may join at corners.

Worked Example

Problem: Determine whether the curve y=x3y = x^3 is smooth on (−∞,∞)(-\infty, \infty).
Step 1: Compute the derivative of f(x)=x3f(x) = x^3.
f′(x)=3x2f'(x) = 3x^2
Step 2: Check whether f′(x)f'(x) exists for all xx. Since 3x23x^2 is a polynomial, it is defined for every real number.
Step 3: Check continuity. Every polynomial is continuous on (−∞,∞)(-\infty, \infty), so f′(x)=3x2f'(x) = 3x^2 is continuous everywhere.
Answer: Yes, y=x3y = x^3 is a smooth curve on (−∞,∞)(-\infty, \infty) because its derivative 3x23x^2 exists and is continuous everywhere.

Why It Matters

Many theorems in calculus — including Green's theorem, Stokes' theorem, and arc length formulas — require the curve to be smooth (or at least piecewise smooth). In physics and engineering, smooth curves model trajectories where velocity changes continuously, which is essential for computing work, flux, and curvature.

Common Mistakes

Mistake: Assuming a continuous curve is automatically smooth.
Correction: Continuity means no breaks, but smoothness also requires a continuous derivative. The curve y=∣x∣y = |x| is continuous everywhere but not smooth at x=0x = 0 because f′(0)f'(0) does not exist (there is a corner).