Foundation lesson
Partial derivatives and held-fixed quantities
A quantity that depends on more than one variable has more than one rate of change. A partial derivative varies one variable and holds the others fixed, and it means nothing until you say which ones are held fixed.
Written for this edition, not translated from Einstein.
When a quantity depends on two things, what does its rate of change mean?
The concentration of particles in a tube depends on where you look, x, and when you look, t. Ask how fast the concentration changes and there are two different answers: how it changes as you move along the tube at one instant, and how it changes at one place as time passes.
A partial derivative picks one of them. ∂f/∂x is the rate of change along the tube with the time t held fixed, like comparing points in one photograph. ∂f/∂t is the rate of change in time at a fixed position x, like watching one spot under the microscope. The curly ∂ says that something else is being held fixed, and a subscript names it:
The partial derivative of f with respect to x at fixed t, and the partial derivative of f with respect to t at fixed x.
Thermodynamics depends on the same care. How much a gas's volume changes as the pressure changes depends on whether its temperature T is held fixed, with heat flowing in and out, or its entropy S is held fixed, with the gas insulated. The two derivatives have different values, and ∂V/∂p written alone does not say which is meant:
The partial derivative of V with respect to p at fixed temperature T, versus the same derivative at fixed entropy S.
In §3 of the light-quanta paper, radiation fills a fixed volume and φ is its entropy per unit volume and per unit interval of frequency, a function of the energy density ρ and of ν. Varying ρ with the frequency ν held fixed, Einstein finds that the derivative is 1/T:
The partial derivative of phi with respect to rho, at fixed frequency nu, equals one over T.
In §6 of the relativity paper, Maxwell's equations are rewritten in the coordinates ξ, η, ζ, τ of a moving system. A derivative with respect to x taken with y, z and t held fixed then becomes a combination of a derivative with respect to ξ and one with respect to τ, because holding the resting time t fixed does not hold the moving time τ fixed.
Worked example: The same diffusion profile, differentiated two ways
Take the spreading profile of the Brownian paper's §4, written for a single particle:
f of x and t equals one over the square root of four pi D t, times e to the minus x squared over four D t.
Hold t fixed and differentiate with respect to x. Only the exponent depends on x, and the derivative of −x²/4Dt with respect to x is −x/2Dt:
The partial derivative of f with respect to x at fixed t equals minus x over two D t, times f.
That is the slope of the profile in one snapshot: downhill to the right of the centre, uphill to the left. Now hold x fixed and differentiate with respect to t. Both the factor in front and the exponent depend on t:
The partial derivative of f with respect to t at fixed x equals minus one over two t, plus x squared over four D t squared, all times f.
That is what a probe fixed at one place records. Near the centre, where x² is less than 2Dt, the bracket is negative: the concentration falls as the cloud spreads away. Farther out it is positive: the concentration rises as the cloud arrives. Differentiate the first result once more with respect to x and multiply by D, and you get the second. That is the §4 equation ∂f/∂t = D ∂²f/∂x², a rate at one place equal to a curvature at one time.
Try it: hold time fixed, or hold position fixed
In §4 of the Brownian paper, f(x, t) is the number of particles per unit volume at a place x along a tube, at a time t. “How fast does the concentration change?” then has two answers, and which one you mean depends on what you keep still: the moment, or the place.
One moment: t fixed, ∂f/∂x
Freeze the moment and compare neighbouring places, as in a single photograph of the tube. ∂f/∂x says how steeply the concentration changes along it: particles per µm³ for each µm, which is particles per µm⁴.
Particles drift down that slope. The flux, the number crossing each square micrometre in a second, is J = −D ∂f/∂x: Fick’s first law.
One place: x fixed, ∂f/∂t
Keep your eye on one spot under the microscope and let time run. ∂f/∂t says how fast the count there rises or falls: particles per µm³ for each second.
Particles are neither made nor destroyed, so the count at a spot changes only by what flows in and out: ∂f/∂t = −∂J/∂x = D ∂²f/∂x². That is the diffusion equation of §4.
The same question in thermodynamics
Squeeze a gas and its pressure rises, but by how much depends on what else you keep fixed:
| Held fixed | Derivative | What happens |
|---|---|---|
| Temperature T (isothermal) | (∂p / ∂V)T | Heat flows in or out to keep the temperature constant. |
| Entropy S: no heat exchanged (adiabatic) | (∂p / ∂V)S | The gas warms as it is squeezed, so its pressure climbs faster than at fixed temperature. |
| Volume V (isochoric) | (∂p / ∂T)V | In a rigid vessel the pressure rises as the gas is heated. |
What it shows, in words
∂f/∂x compares places at one moment; ∂f/∂t follows one place through time. They answer different questions and carry different units. In thermodynamics a subscript names what is held fixed, so (∂p/∂V)T and (∂p/∂V)S are different numbers for the same gas.
Where this lesson stops
A partial derivative is not defined until you have said which quantities are held fixed.
If you came here from a passage, Back returns you to the exact place you left.