Vector-valued functions of one real variable
When a function returns a vector instead of a number, it describes a motion. Differentiating it gives the velocity; differentiating again gives the acceleration, and almost all of kinematics follows from there.
01A function that returns a vector
So far, functions have taken a number and returned a number. A vector-valued function of one real variable takes a number —typically time— and returns a vector: the position of something that is moving.
| Curve | Vector equation | Where it appears |
|---|---|---|
| Line | \( r(t) = r_{0} + t\cdot d \) | The one from line and plane, now read as a motion |
| Circle | (R cos t, R sin t) | A rotating phasor: the representation of alternating current |
| Helix | \( (R \cos t, R \operatorname{sin} t, c\cdot t) \) | Path of an electron in a magnetic field; a winding |
| Lissajous | \( (A \operatorname{sin} (at), B \operatorname{sin} (bt + \varphi )) \) | What an oscilloscope draws in XY mode |
02Differentiating: everything component by component
The limit, the continuity and the derivative of a vector-valued function are computed in each component separately. That means everything learned in Calculus I still holds:
The differentiation rules carry over with one important caveat: there are two products. For the scalar one, (u · v)′ = u′ · v + u · v′; for the vector one, (u × v)′ = u′ × v + u × v′, and here the order cannot be changed, because the cross product is not commutative.
If |r(t)| is constant —the moving point travels on a sphere or a circle—, then r · r′ = 0: the velocity is perpendicular to the position. This is what explains why, in uniform circular motion, the acceleration points to the center, and why an electron in a magnetic field turns without changing its speed: the force is always perpendicular to the velocity and does no work.
03Arc length
When the curve is parametrized by the distance traveled along it, the arc-length parameter s, the speed equals 1 and the formulas simplify. This is the “natural” parametrization: traveling along the curve at unit constant speed.
04How a path bends
At each point of the curve, three mutually perpendicular vectors are defined, the Frenet frame:
- Unit tangent T = r′/|r′|: where it is heading.
- Principal normal N = T′/|T′|: where it bends toward, always pointing to the concave side.
- Binormal B = T × N: perpendicular to the plane in which the curve is bending.
You move the parameter and see the tangent vector, the normal vector, the speed and the circle that best fits the curve at that point.
05Where they appear in electronics and physics
- The phasor. A sinusoidal voltage is represented by r(t) = V(cos ωt, sin ωt): a vector-valued function whose magnitude is constant and whose derivative, perpendicular to it, leads by 90°. That perpendicularity is the phase shift between voltage and current in a capacitor. It is seen in the vector study of alternating current.
- Charge in a magnetic field. With a velocity not parallel to the field, the path is a helix: a circle in the plane perpendicular to B and uniform advance along B. This is the principle of the cyclotron and of cathode-ray tubes.
- Lissajous figures. Two signals on the X and Y axes of the oscilloscope form a parametric curve; the frequency ratio determines the figure and the phase shift determines its tilt.
- Robotics and CNC. The path of a tool is r(t), and the controller needs bounded velocity and acceleration: that is why corners are rounded, since an infinite curvature would require infinite acceleration.
06In the lab
With two generators in XY mode, obtain the figures for 1:1, 1:2 and 3:2 ratios, and measure the phase shift from the tilt of the ellipse. Write the vector-valued function of each figure. You can try it first in the instrument simulator.
Compute the length of one arc of a helix of radius 5 cm and pitch 2 cm, for one complete turn, and check it by measuring the wire needed to wind it.
For a moving point that travels along a curve with increasing speed, compute the tangential and normal components of the acceleration at three points and verify that their vector sum equals r″.
07Common mistakes
- Confusing the curve with the function: the same curve admits infinitely many parametrizations, which traverse it at different speeds.
- Differentiating the magnitude instead of the vector: |r|′ is not |r′|.
- Changing the order in (u × v)′.
- Forgetting to normalize when computing T and N.
- Believing that zero acceleration implies constant velocity in direction: in uniform circular motion the speed is constant but the acceleration is not zero.
08Self-assessment
For r(t) = (cos t, sin t, t), find v and its magnitude.
v = (−sin t, cos t, 1) and |v| = √(sin² + cos² + 1) = √2, constant: the helix is traversed at uniform speed.
What is the length of one turn of that helix?
L = ∫₀2π √2 dt = 2π√2 ≈ 8.89.
Why is r · v = 0 in uniform circular motion?
Because |r| is constant: differentiating r · r = const gives 2 r · r′ = 0.
What does curvature measure?
How much the curve bends per unit of length. Its reciprocal is the radius of the osculating circle: line → κ = 0; circle of radius R → κ = 1/R.
A moving point speeds up while turning. How is its acceleration divided?
Into a tangential component, d|v|/dt, which changes the speed, and a normal one, κ|v|², which changes the direction.
09Further reading
- James Stewart. Multivariable Calculus. 7th ed. (Spanish edition, “Cálculo de varias variables. Trascendentes tempranas”), Cengage Learning, 2012. Chapter 13 develops vector functions, arc length and curvature with good figures.
- Jerrold E. Marsden and Anthony J. Tromba. Vector Calculus. 5th ed. (Spanish edition, “Cálculo vectorial”), W. H. Freeman, 2003. The reference text for the course, more demanding and with excellent physical applications.
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. The motion of charges in magnetic fields, which is the central physical example of this topic.