Scalar functions of several variables
The temperature of a plate, the potential around two charges or the gain of an amplifier as a function of two components: all of them are functions of several variables.
01One output, several inputs
The temperature of a plate depends on where it is measured: T(x, y). The power dissipated by a resistor depends on the voltage and the resistance: P(V, R). The gain of a stage depends on two components. All of these are scalar functions of several variables: two or more numbers go in, one comes out.
The domain is no longer an interval but a region of the plane: the inside of a circle, a half-plane, the plane minus a line. The restrictions are the same as always —no dividing by zero, no even roots of negative numbers, no logarithms of non-positive numbers—, only now they define a zone.
For f(x, y) = √(9 − x² − y²), we need 9 − x² − y² ≥ 0, that is, x² + y² ≤ 9: the domain is the closed disk of radius 3. The surface is the upper hemisphere of radius 3.
02Level curves: the map of the surface
Drawing surfaces on paper is hard; that is why they are sliced. A level curve is the set of points where f takes a constant value: f(x, y) = k. It is exactly what a topographic map does with the height of the terrain.
| On the map | On the surface | In physics |
|---|---|---|
| Curves very close together | Steep slope | Strong field: the gradient is large |
| Curves far apart | Nearly flat area | Weak field |
| Closed curves that shrink | A peak or a pit | Local maximum or minimum |
| X-shaped curves | A saddle point | Unstable equilibrium |
The level curves of an electric potential are the equipotentials, and the field lines cross them perpendicularly. That perpendicularity is no coincidence: it is proved in the differential calculus of several variables, when the gradient appears.
The color is the value of the function and the lines are the level curves. Tap the map to read the value at a point.
03The surfaces worth recognizing
| Equation | Surface | Level curves |
|---|---|---|
| \( z = ax + by + c \) | Plane | Parallel lines |
| \( z = x^{2} + y^{2} \) | Paraboloid (a bowl) | Concentric circles |
| \( z = x^{2} - y^{2} \) | Hyperbolic paraboloid: the saddle | Hyperbolas |
| \( x^{2} + y^{2} + z^{2} = R^{2} \) | Sphere | Circles |
| \( x^{2} + y^{2} = R^{2} \) | Cylinder (z free) | One circle repeated |
These are the same quadrics that appeared in quadratic forms, and their classification again depends on the signs of the eigenvalues. With three variables a graph is no longer possible: we work with level surfaces, f(x, y, z) = k, which for an electric potential are the equipotential surfaces.
04Limits: approaching from every side
Here is the most important difference from Calculus I. In one variable, a point is approached from the left or from the right: two paths. In two variables there are infinitely many paths, and the limit exists only if all of them give the same result.
For f(x, y) = xy/(x² + y²) at the origin: approaching along the x axis (y = 0) the value is always 0; approaching along the line y = x it becomes x²/(2x²) = 1/2.
Two paths, two results: the limit does not exist, even though both iterated limits are 0. Finding two paths with different results is the standard way to prove that a limit does not exist; to prove that it does exist you have to bound the function, for example by switching to polar coordinates.
In polar coordinates, with x = ρcos θ and y = ρsin θ, the limit as (x, y) → (0, 0) becomes a limit as ρ → 0. If the result depends on θ, there is no limit; if it does not and the bound is uniform, there is.
Continuity is defined as before: f is continuous at a point if the limit exists and equals the value. And the theorems that matter still hold: sums, products, quotients and compositions of continuous functions are continuous, and a continuous function on a closed, bounded region attains a maximum and a minimum.
05Where they appear in electronics
- Electric potential. V(x, y, z) is a scalar function of three variables; its level surfaces are the equipotentials and its gradient, with the sign changed, is the electric field.
- Thermal maps. The temperature of a circuit board is T(x, y); the level curves show where the hot spot is, which is what a thermal imaging camera sees.
- Response surfaces. The gain of a stage as a function of two resistors, or the efficiency of a power supply as a function of output voltage and current: optimizing means looking for the maximum of a surface.
- Characteristic curves. The collector current of a transistor depends on two variables, IC(VCE, IB): the family of curves in the datasheet is really a slice of a surface. See bipolar transistors.
06In the lab
Determine and sketch the domain of f(x,y) = ln(x + y), of g(x,y) = 1/√(4 − x² − y²) and of h(x,y) = arcsin(x + y). In each case, state whether it is open, closed and bounded.
Using an infrared thermometer, measure the temperature of a board with one component dissipating power, on a 5 × 5 grid of points, and draw the level curves by hand. Compare with what the lab shows for the Gaussian bell.
For f(x,y) = (x²y)/(x⁴ + y²), compute the limit at the origin along the lines y = mx and then along the parabola y = x². Check that along every line it gives 0 and along the parabola it gives 1/2: lines are not enough.
07Common mistakes
- Concluding that the limit exists because it agrees along several lines. All paths are needed, and lines are only a few of them.
- Confusing iterated limits with the double limit: both iterated limits can exist, be equal, and the limit can still fail to exist.
- Describing the domain with an equation instead of a region.
- Drawing level curves without indicating which value each one corresponds to.
- Forgetting that level curves never cross: if they did, a point would have two values.
08Self-assessment
Domain of f(x, y) = ln(9 − x² − y²).
9 − x² − y² > 0, that is, x² + y² < 9: the inside of the circle of radius 3, without the boundary.
What are the level curves of z = x² + y²?
Concentric circles centered at the origin: for z = k, of radius √k, with k ≥ 0.
Does the limit of xy/(x² + y²) exist at the origin?
No: along the x axis it gives 0 and along the line y = x it gives 1/2.
What does a level curve of the electric potential represent physically?
An equipotential: moving a charge along it costs no work, because the field is perpendicular to the curve.
If the level curves are very close together, what happens to the function?
It varies quickly in that zone: the slope —and the magnitude of the gradient— is large.
09Further reading
- James Stewart. Multivariable Calculus. 7th ed. (Spanish edition, “Cálculo de varias variables. Trascendentes tempranas”), Cengage Learning, 2012. Chapter 14 opens with domains, level curves and limits, with many surfaces drawn out.
- Jerrold E. Marsden and Anthony J. Tromba. Vector Calculus. 5th ed. (Spanish edition, “Cálculo vectorial”), W. H. Freeman, 2003. More rigorous on limits in several variables and on the topology of the domain.
- Claudio Pita Ruiz. Cálculo vectorial. Prentice Hall (in Spanish). Written in Spanish and very detailed in its examples of limits that do not exist.