Calculus on Curves

A parametric curve is a function $\mathbf{r}:[a,b]\to\mathbb{R}^3$ assigning to each value of the parameter $t$ a point $\mathbf{r}(t)=(x(t),y(t),z(t))$. The parameter $t$ is often time: $\mathbf{r}(t)$ is position, $\mathbf{r}'(t)$ velocity, $\mathbf{r}''(t)$ acceleration.

Curve vs trace. The curve is the function $\mathbf{r}$, not just the set of points it describes (the trace). The same trace can be traversed at different speeds, in different directions, several times: these are distinct parametrisations. Example: $\mathbf{r}_1(t)=(\cos t,\sin t)$ and $\mathbf{r}_2(t)=(\cos 2t,\sin 2t)$ draw the same circle, but the second runs twice as fast.

Regular curve. A curve is regular if $\mathbf{r}'(t)\neq\mathbf{0}$ for all $t$: the velocity never vanishes. This guarantees a well-defined direction of travel, the unit tangent:

Where $\mathbf{r}'=\mathbf{0}$ the curve may have cusps or corners (e.g. the cycloid touches the ground with zero velocity and forms a cusp).

How do you measure the length of a crooked curve? Idea: approximate it with many tiny segments, sum their lengths, take the limit. In a small interval $dt$ the point moves by $\mathbf{r}'(t)\,dt$, a distance $|\mathbf{r}'(t)|\,dt$. Summing (integrating):

The quantity $ds=|\mathbf{r}'(t)|\,dt$ is the arc-length element: the "infinitesimal ruler" weighting line integrals.

Independence of parametrisation. Length is a geometric property of the trace, not of how we traverse it: a faster parametrisation has larger $|\mathbf{r}'|$ but a shorter time interval, and the two effects cancel exactly.

In polar coordinates. If the curve is $r=r(\theta)$, setting $x=r\cos\theta$, $y=r\sin\theta$ and computing gives

The $r$ term accounts for angular displacement (arc $r\,d\theta$), the $r'$ term for radial displacement.

Arc-length parameter. One can reparametrise by $s$ (distance travelled) so that $|\mathbf{r}'(s)|=1$: the curve is then traversed at unit speed, and many formulas (curvature, Frenet) simplify.

Curvature $\kappa$ measures how fast the curve changes direction: the rate of rotation of the unit tangent with respect to distance, $\kappa=|d\hat T/ds|$. A line has $\kappa=0$ (direction never changes); a circle of radius $R$ has $\kappa=1/R$ constant (smaller circle, more curved).

For a general space curve, the practical formula uses velocity and acceleration:

Why the cross product? Only the component of acceleration perpendicular to velocity bends the path (the parallel part only changes the speed). The product $\mathbf{r}'\times\mathbf{r}''$ isolates exactly that perpendicular component, and its magnitude measures it.

For a graph $y=f(x)$, taking $\mathbf{r}(x)=(x,f(x))$:

Osculating circle. At each point, the radius of curvature is $\rho=1/\kappa$: the radius of the circle that best approximates the curve there (it "kisses" it, from osculum). A car on a curve of radius $\rho$ undergoes centripetal acceleration $v^2/\rho$: that is why tight curves (small $\rho$, large $\kappa$) must be taken slowly.

We want to "sum" the values of a scalar function $f(x,y,z)$ along a curve $\gamma$. Weighting each value by the arc-length element $ds$:

Physical meaning. If $\gamma$ is a wire and $f$ its linear density (mass per unit length), then $\int_\gamma f\,ds$ is the wire's total mass. With $f=1$ we recover the length. The average value of $f$ along $\gamma$ is $\frac{1}{L}\int_\gamma f\,ds$.

Orientation independence. Since $ds=|\mathbf{r}'|\,dt\geq 0$, the 1st-kind integral does not change if the curve is traversed in the opposite direction: the mass of a wire does not depend on which end you call the "start".

For a vector field $\mathbf{F}=(P,Q,R)$ (e.g. a force, an electric field) we care about the component tangent to the curve: that is what does work. The line integral of the 2nd kind is

Physical meaning. If $\mathbf{F}$ is a force, this is the work done along $\gamma$. The dot product $\mathbf{F}\cdot d\mathbf{r}$ projects the force onto the direction of motion: a force perpendicular to the path does no work (like the centripetal force on a circular orbit).

Orientation dependence. Unlike the 1st kind, here the sign flips when the direction is reversed: $\int_{-\gamma}\mathbf{F}\cdot d\mathbf{r}=-\int_{\gamma}\mathbf{F}\cdot d\mathbf{r}$. Going back along the same path returns the work done on the way out.

A field $\mathbf{F}$ is conservative if it is the gradient of a scalar function $U$, called the potential: $\mathbf{F}=\nabla U$. For these fields something magical happens: the work depends not on the path but only on the endpoints.

Immediate consequence: on a closed path ($A=B$) the work is zero, $\oint_\gamma\mathbf{F}\cdot d\mathbf{r}=0$. This is energy conservation: gravity or the electrostatic force neither "create nor destroy" energy over a full loop.

How to recognise a conservative field. In $\mathbb{R}^2$, on a simply connected domain (no holes),

This is the same exactness condition as for ODEs: the cross derivatives must agree (symmetry of the Jacobian). Mind the domain: on domains with holes, $\partial_y P=\partial_x Q$ is necessary but not sufficient (classic counterexample: the "vortex" field around the origin).

Finding the potential: integrate $U_x=P$ in $x$, then impose $U_y=Q$ to fix the integration constant — exactly as for exact equations.

Many curves are naturally described in polar coordinates $r=r(\theta)$, where $r$ is the distance from the origin and $\theta$ the angle. The link to Cartesian coordinates is $x=r\cos\theta$, $y=r\sin\theta$.

• Circle: $r=R$ (constant).
• Archimedean spiral: $r=a\theta$ — the radius grows linearly with the angle (record grooves, spiral springs).
• Cardioid: $r=a(1+\cos\theta)$ — heart shape, polar pattern of a directional microphone.
• Rose: $r=a\cos(k\theta)$ — petals ($k$ petals if $k$ odd, $2k$ if even).

The polar velocity has a radial component $r'$ and a transverse component $r$, giving the arc-length element $ds=\sqrt{r^2+r'^2}\,d\theta$ seen earlier. The area swept by a polar curve is $A=\frac12\int_\alpha^\beta r^2\,d\theta$ — the formula behind Kepler's second law (the Sun–planet radius sweeps equal areas in equal times).

At each point of a regular space curve one builds a moving reference frame, the Frenet frame $(\hat T,\hat N,\hat B)$, that accompanies the motion:

• Unit tangent $\hat T=\mathbf{r}'/|\mathbf{r}'|$: direction of motion.
• Unit normal $\hat N=\hat T'/|\hat T'|$: points toward the centre of curvature (where the curve bends).
• Unit binormal $\hat B=\hat T\times\hat N$: perpendicular to the osculating plane.

The Frenet formulas link the derivatives of these vectors to two geometric quantities: curvature $\kappa$ (how much the curve bends in the osculating plane) and torsion $\tau$ (how much it leaves that plane, "twisting"):

A plane curve has $\tau=0$; a helix has constant $\kappa$ and $\tau$. Curvature and torsion determine the curve up to rigid motions (fundamental theorem of curves).