Differentials of Multivariable Functions
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 4.4: “Tangent Planes and Linear Approximations” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/4-4-tangent-planes-and-linear-approximations |
| Textbook used in class | Stewart, Calculus, Section 14.4: “Tangent Planes and Linear Approximations” (Examples 5, 6) |
Quick Reference
Total differential of $z = f(x,y)$: $$dz = f_x(x,y)\,dx + f_y(x,y)\,dy = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy.$$
Error estimation: $\Delta z \approx dz$ when $\Delta x = dx$ and $\Delta y = dy$ are small.
For $f(x,y,z)$: $dw = f_x\,dx + f_y\,dy + f_z\,dz$.
Motivation
A physical measurement $z = f(x,y)$ depends on two measured quantities $x$ and $y$. Each measurement has a small error: $x$ is measured to within $\pm\,dx$, and $y$ to within $\pm\,dy$. The total differential $dz$ gives the worst-case combined error in $z$ due to both measurement uncertainties. This is the standard tool for error propagation in laboratory science and engineering.
The differential $dz = f_x\,dx + f_y\,dy$ is simply the numerator in the linearization: $dz$ is the change in $z$ along the tangent plane when $x$ changes by $dx$ and $y$ changes by $dy$.
Key Concepts
1. The Total Differential
The differential $dz$ is defined as $$dz = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy.$$
Here $dx$ and $dy$ are formal differentials representing small changes in $x$ and $y$. The relationship $\Delta z \approx dz$ says that the actual change in $z$ is approximately the differential when the increments are small.
2. Connection to Linearization
The linearization says $\Delta z = f(a+\Delta x, b+\Delta y) - f(a,b) \approx f_x(a,b)\,\Delta x + f_y(a,b)\,\Delta y$. Setting $dx = \Delta x$ and $dy = \Delta y$, this is exactly $dz$. The differential and the linearization increment are the same object.
3. Error Bounds
If $|dx| \leq e_1$ and $|dy| \leq e_2$, then the maximum error $|dz|$ is bounded by $|f_x|\,e_1 + |f_y|\,e_2$ (using the triangle inequality). This gives a conservative bound on the propagated error.
Worked Example
The dimensions of a rectangular box are measured as $l = 60$, $w = 40$, $h = 20$ cm, each with an error of at most 0.2 cm. Estimate the maximum error in the computed volume. (Adapted from Stewart 14.4, Example 6.)
Volume: $V = lwh$.
Total differential: $$dV = wh\,dl + lh\,dw + lw\,dh.$$
At $(l, w, h) = (60, 40, 20)$: coefficients are $wh = 800$, $lh = 1200$, $lw = 2400$.
Maximum error: $|dV| \leq 800(0.2) + 1200(0.2) + 2400(0.2) = 160 + 240 + 480 = 880$ cm$^3$.
So the computed volume of $V = 48{,}000$ cm$^3$ has an error of at most $880$ cm$^3$, or about 1.8%.
$dz$ gives the exact change in $z$, not just an approximation. The differential $dz$ is the change along the tangent plane, not along the surface. The actual change $\Delta z$ differs from $dz$ by a small amount that becomes negligible as $dx, dy \to 0$. For finite (non-infinitesimal) $\Delta x, \Delta y$, the differential is an approximation, and the quality of the approximation depends on how curved the surface is.
Leveled Practice
Problem 1. The radius and height of a cylinder are measured as $r = 5$ cm (error $\pm 0.05$) and $h = 20$ cm (error $\pm 0.1$). Estimate the maximum error in the volume $V = \pi r^2 h$.
Show answer
$dV = 2\pi r h\,dr + \pi r^2\,dh$.
At $(r,h) = (5,20)$: $2\pi(5)(20) = 200\pi$, $\pi(25) = 25\pi$.
$|dV| \leq 200\pi(0.05) + 25\pi(0.1) = 10\pi + 2.5\pi = 12.5\pi \approx 39.3$ cm$^3$.