Case Study 1 — Reading a Weather Map: Temperature and Pressure Fields
Field: Meteorology and physics (atmospheric science)
Every weather map you have ever seen is a contour plot of a function of several variables. When a meteorologist points at the swirling lines on the evening forecast, she is reading level curves — the exact objects of Section 29.4 — and the local rate at which temperature and pressure change is a partial derivative. This case study follows an atmospheric scientist, Dr. Lena Ortiz, through one morning's analysis of a developing weather system over a flat coastal plain, and shows how the machinery of Chapter 29 turns a grid of numbers into a forecast.
The setting: a field, not a number
Lena's weather station network covers a region she models with coordinates $(x, y)$ in kilometers east and north of the station at the origin. At 6:00 a.m. the surface temperature field (in degrees Celsius) is well approximated by
$$T(x, y) = 18 + \frac{x^2}{20} - \frac{y^2}{40} + \frac{xy}{50}.$$
This is not a temperature; it is a rule that assigns a temperature to every point — a function $T: \mathbb{R}^2 \to \mathbb{R}$ exactly as defined in Section 29.1. The single most important mental shift, the one the whole chapter is built around, is already here: the weather over the plain is not described by one number but by a surface floating above the map, with altitude $z = T(x, y)$ standing for warmth.
The domain is all of the modeled plain (no square roots or denominators to forbid anything, by the logic of Section 29.2), and the range is whatever band of temperatures the formula attains over the region Lena cares about. Near the origin the constant term tells her the baseline is about $18\,^\circ\text{C}$; the quadratic terms describe how warmth rises and falls as she moves away.
The contour map: isotherms
Lena never plots the temperature surface in 3D — no forecaster does. Instead she draws isotherms, the level curves $T(x, y) = c$ where the temperature equals a fixed value (Section 29.4). On a weather map these are exactly the lines of equal temperature, and their geometry tells her where fronts are forming.
Consider the isotherm $T = 18$. Setting the formula equal to $18$ and clearing the constant gives
$$\frac{x^2}{20} - \frac{y^2}{40} + \frac{xy}{50} = 0.$$
Because the $x^2$ and $y^2$ terms carry opposite signs, this level curve is a hyperbola-like cross through the origin — the signature pattern Section 29.4 flagged as the fingerprint of a saddle. And that is precisely what the temperature field has at the origin: walk east and it warms ($x^2/20$ grows), walk north and it cools ($-y^2/40$ falls). The same point is a low point along one heading and a high point along another. Lena recognizes this instantly as a col — the meteorologist's name, borrowed from mountaineering, for a saddle in a pressure or temperature field, a neutral point between warm and cool air masses.
The spacing of the isotherms carries the rest of the story. Where Lena draws isotherms at every $2\,^\circ\text{C}$ and finds them bunched tightly, the temperature is changing fast over a short distance — a thermal gradient, the leading edge of a front. Where the isotherms spread far apart, the air is thermally uniform. She has read the steepness of the surface without ever lifting it into three dimensions, exactly as Section 29.4 promised: contour spacing, not contour count, encodes the rate of change.
Partial derivatives: how fast, in which direction
To quantify the front, Lena needs the rate at which temperature changes per kilometer. But in which direction? This is the question that forces the move from one derivative to partial derivatives (Section 29.8). She computes both:
$$\frac{\partial T}{\partial x} = \frac{2x}{20} + \frac{y}{50} = \frac{x}{10} + \frac{y}{50}, \qquad \frac{\partial T}{\partial y} = -\frac{2y}{40} + \frac{x}{50} = -\frac{y}{20} + \frac{x}{50}.$$
Each is found by the chapter's one rule: differentiate one variable, freeze the rest. For $T_x$ she treats $y$ as a constant, so the $-y^2/40$ term vanishes and the $xy/50$ term contributes $y/50$. For $T_y$ she treats $x$ as constant, so $x^2/20$ vanishes and $xy/50$ contributes $x/50$.
At the monitoring point $(x, y) = (10, 20)$ — ten kilometers east, twenty north — she evaluates:
$$T_x(10, 20) = \frac{10}{10} + \frac{20}{50} = 1 + 0.4 = 1.4\ \tfrac{^\circ\text{C}}{\text{km}},$$ $$T_y(10, 20) = -\frac{20}{20} + \frac{10}{50} = -1 + 0.2 = -0.8\ \tfrac{^\circ\text{C}}{\text{km}}.$$
Now she can speak precisely. Stepping due east from $(10, 20)$, the temperature rises about $1.4\,^\circ\text{C}$ per kilometer; stepping due north, it falls about $0.8\,^\circ\text{C}$ per kilometer. The geometric reading from Section 29.8 is exactly the hiker's: $T_x$ is the east–west slope of the temperature "landscape," $T_y$ the north–south slope. These are two of the infinitely many directional slopes — the rest wait for the gradient and directional derivative of Chapter 30, which will tell Lena the single compass heading of fastest warming. For this morning, the two partials are enough.
The pressure field and the ideal-gas connection
Temperature is only half the forecast. Lena overlays a pressure field $P(x, y)$, whose level curves are the isobars — the closed loops a weather map draws around high- and low-pressure centers. A nested set of closed isobars with no crossing marks a pressure extremum: a tight cluster of small loops is a low (a pit in the pressure surface), and the tighter they pack, the stronger the pressure gradient and the faster the wind. This is the contour-reading rule of Section 29.4 applied to a second field — nested closed loops mark a peak or pit, and spacing measures steepness.
The physics linking the two fields is the ideal gas law from Section 29.13, $P = nRT/V$, itself a function of several variables. Its partials,
$$\frac{\partial P}{\partial V} = -\frac{nRT}{V^2} < 0, \qquad \frac{\partial P}{\partial T} = \frac{nR}{V} > 0,$$
tell Lena that, at fixed volume, warming a parcel of air raises its pressure ($\partial P/\partial T > 0$), while expansion lowers it ($\partial P/\partial V < 0$). The signs alone — no numbers needed — explain why the warm region she identified from the temperature field tends to sit over rising, lower-density air. The level surfaces $P = c$ in $(V, T)$ space are the isobaric curves of any thermodynamics text, and the same word "level surface" (Section 29.5) describes the isobars on her map and the constant-pressure surfaces aloft.
Putting it together: a short-range forecast by linearization
By 6:15 a.m. Lena wants a quick estimate of the temperature at a point $0.5\text{ km}$ east and $0.3\text{ km}$ north of her monitoring station, without re-evaluating the full formula. This is exactly what the tangent plane and linearization of Section 29.10 are for. The linearization of $T$ at the base point $(10, 20)$ is
$$L(x, y) = T(10, 20) + T_x(10, 20)\,(x - 10) + T_y(10, 20)\,(y - 20).$$
First the base value: $T(10, 20) = 18 + \dfrac{100}{20} - \dfrac{400}{40} + \dfrac{200}{50} = 18 + 5 - 10 + 4 = 17\,^\circ\text{C}$. Then, with $\Delta x = 0.5$ and $\Delta y = 0.3$,
$$L = 17 + 1.4(0.5) + (-0.8)(0.3) = 17 + 0.7 - 0.24 = 17.46\,^\circ\text{C}.$$
The linear estimate is $17.46\,^\circ\text{C}$. The exact value, $T(10.5, 20.3)$, works out to $17.473\,^\circ\text{C}$ — the linearization is off by only about a hundredth of a degree, because the step is small and the discarded error grows like the square of the distance from the base point (Section 29.10). Step a full kilometer away and the approximation degrades fast; the Common Pitfall of Section 29.10 — never trust a linearization far from its base point — is a working forecaster's daily discipline. For a fifteen-minute, short-range nowcast over half a kilometer, the tangent plane is exactly the right tool, and it is the same "approximation with precision" that Chapter 11 introduced for a single variable, now lifted to a surface.
Discussion Questions
- Lena identified a col (saddle) in the temperature field at the origin. Using the partials $T_x$ and $T_y$, verify that both vanish at $(0,0)$. Why does $T_x = T_y = 0$ fail to tell her whether the origin is a peak, a pit, or a saddle, and which chapter supplies the test that settles it?
- The isotherm spacing tells Lena where the thermal gradient is strongest. In terms of contour spacing, where on the map would you expect the largest values of $|T_x|$ and $|T_y|$ — and how does this preview the gradient of Chapter 30?
- The ideal-gas partials $\partial P/\partial V$ and $\partial P/\partial T$ had definite signs but the case study never plugged in numbers. Argue why, for a forecaster, the sign of a partial derivative is often more important than its magnitude.
- Suppose the temperature step Lena needed were $5$ km east instead of $0.5$ km. Estimate qualitatively how much worse the linearization would get, citing the squared-distance error rule.
Annotated Reading
- Stewart, Calculus: Early Transcendentals, §14.1 (Functions of Several Variables) and §14.3 (Partial Derivatives). Stewart's weather-map and heat-index examples are the direct classroom ancestor of this case study; §14.1 introduces level curves with topographic and isothermal maps.
- OpenStax Calculus Volume 3, §4.1–4.3. Free and parallel to the above; §4.1's contour-map figures and §4.3's partial-derivative interpretations map one-to-one onto Lena's analysis.
- Wallace & Hobbs, Atmospheric Science: An Introductory Survey (2nd ed.), Ch. 1 and Ch. 7. The professional vocabulary — isotherms, isobars, cols, thermal gradients — defined and used by working meteorologists; shows that "level curve" and "weather map" are the same object.