# Partial derivatives and held-fixed quantities

Authored explanation; editorial review pending.

Many physical quantities depend on more than one parameter: particle density depends on both position x and time t; gas entropy depends on both volume V and temperature T. When asking how such a quantity changes, one must specify which variable is moving and which variables are being held constant.

The partial derivative ∂f/∂x represents the rate of change of f with respect to x while holding time t strictly fixed. Conversely, ∂f/∂t represents the rate of accumulation at a fixed position x over time. In thermodynamics, holding temperature fixed produces an isothermal derivative, while holding entropy or volume fixed produces an adiabatic or isochoric derivative.

Einstein's derivation of the diffusion equation equates the time rate of accumulation at a fixed location to the divergence of spatial flux: ∂f/∂t = D ∂^2f/∂x^2. Both sides describe rates under different held-fixed constraints.

## Worked example

$$
\left(\frac{\partial f}{\partial x}\right)_t = -\frac{x}{2Dt} f(x, t)
$$

Spatial partial derivative of density with respect to x holding time t fixed.

Example 1 (Time fixed when moving through space): For diffusion profile f(x, t) = (4 pi D t)^(-1/2) exp(-x^2 / (4Dt)), the spatial gradient quantifies concentration variation along a channel. Here, elapsed time t is strictly fixed as the held-fixed parameter. A spatial snapshot taken at one fixed instant evaluates to ∂f/∂x = -x/(2Dt) f(x, t).

$$
\left(\frac{\partial f}{\partial t}\right)_x = \left(-\frac{1}{2t} + \frac{x^2}{4Dt^2}\right) f(x, t)
$$

Time partial derivative of density with respect to time holding position x fixed.

Example 2 (Position fixed when tracking time): The time accumulation rate records concentration changes at a fixed location. Here, spatial position x is strictly fixed as the held-fixed parameter. A stationary probe at one coordinate records accumulation rate ∂f/∂t = (-1/(2t) + x^2/(4Dt^2)) f(x, t). Equating accumulation to spatial flux divergence yields ∂f/∂t = D ∂^2f/∂x^2.

$$
\left(\frac{\partial V}{\partial p}\right)_T \quad\text{versus}\quad \left(\frac{\partial V}{\partial p}\right)_S
$$

Isothermal volume derivative with respect to pressure holding temperature T fixed versus adiabatic volume derivative holding entropy S fixed.

Example 3 (Thermodynamic derivatives: isothermal versus adiabatic): In gas thermodynamics, the volume response to pressure depends on thermal boundary conditions. The isothermal derivative (∂V/∂p)_T explicitly names temperature T as the held-fixed quantity while heat exchanges freely with a bath. Conversely, the adiabatic derivative (∂V/∂p)_S explicitly names entropy S as the held-fixed quantity under thermal insulation. Because isothermal compression permits heat release, the isothermal compressibility exceeds the adiabatic compressibility. Writing ∂V/∂p without identifying the held-fixed parameter is physically incomplete.

$$
\left(\frac{\partial s_\nu}{\partial u_\nu}\right)_{V, \nu} = \frac{1}{T}
$$

Monochromatic radiation entropy density derivative with respect to energy density holding volume V and frequency nu fixed.

Example 4 (Radiation entropy derivative in Light Quanta §3): For blackbody radiation at frequency nu, the entropy density s_nu depends on both spectral energy density u_nu and enclosure volume V. Einstein's derivation of Wien's displacement law takes the partial derivative (∂s_nu/∂u_nu)_(V, nu) = 1/T. Here both cavity volume V and radiation frequency nu are strictly held fixed as the held-fixed parameters while varying energy density u_nu.

$$
\left(\frac{\partial}{\partial x}\right)_{y, z, t} = \gamma \left(\frac{\partial}{\partial x'}\right)_{y', z', t'} - \gamma \frac{v}{c^2} \left(\frac{\partial}{\partial t'}\right)_{x', y', z'}
$$

Chain rule transformation of spatial partial derivative holding rest-frame coordinates fixed into moving-frame partial derivatives.

Example 5 (Transformed derivatives in Relativity §6): In coordinate transformations between a resting frame (x, y, z, t) and a moving frame (x', y', z', t'), the partial derivative (∂/∂x)_(y, z, t) explicitly holds resting coordinates y, z, and time t fixed. When expressed in moving coordinates via the relativistic chain rule, it becomes a combination of moving spatial derivative (∂/∂x')_(y', z', t') holding y', z', and t' fixed, and moving time derivative (∂/∂t')_(x', y', z') holding x', y', and z' fixed. Changing coordinate systems changes which physical quantities are held fixed.

A partial derivative is mathematically and physically undefined until every held-fixed parameter is explicitly identified.