Difficulty: Intermediate
Prerequisites: Single-variable calculus (derivatives, graphs of functions), familiarity with equations of circles, ellipses, and hyperbolae in 2D.
Up to now, calculus has dealt with functions of a single variable. This chapter extends everything to functions of two (or more) variables, where the input is a point (x, y) and the output is a single number z = f(x, y). The central new skill is visualising these surfaces without being able to draw them easily: contour maps (collections of level curves) are the main tool. If you can read and draw contour maps fluently, partial derivatives and tangent planes in the next sections will make much more sense. Think of contour maps the way you think of topographic maps: each curve is a slice of the surface at a constant height.
A function of two variables takes a point (x, y) and returns a height z. Level curves are the curves you get by slicing the surface at constant z-values. Contour maps collect several level curves on one xy-plane, and reading them is the single most important visualisation skill in multivariable calculus.
Function of two variables, f(x, y)
A rule that assigns a single real number z to each ordered pair (x, y) in some subset of the xy-plane. Think of it as a height at every point on a flat map.
Domain
The set of all (x, y) pairs for which the function is defined. In simple terms, it is the region of the xy-plane where the formula makes sense (no square roots of negatives, no division by zero, etc.).
Range
The set of all output values z = f(x, y) as (x, y) varies over the domain. This tells you how high and how low the surface reaches.
Graph of f(x, y)
The surface in three-dimensional space consisting of all points (x, y, z) where z = f(x, y). Think of it as the shape you would build if you placed a column of height z above every point (x, y).
Level curve
The curve in the xy-plane satisfying f(x, y) = k for a fixed constant k. In simple terms, it is a horizontal slice through the surface at height k, projected down onto the map. Also called an iso-curve or contour line.
Contour map (contour plot)
A collection of level curves for several values of k, drawn together on the xy-plane. Closely spaced curves mean the surface is steep; widely spaced curves mean it is flat. This is exactly how topographic maps work.
Set f(x, y) = k for several values of k (choose evenly spaced values where possible).
Rearrange each equation into a recognisable curve in the xy-plane (circle, ellipse, line, hyperbola, etc.).
Plot those curves on the same axes and label each with its k-value.
Note spacing: where curves crowd together the surface is steep; where they spread out, the surface is nearly flat.
Domain: 9 - 9x^2 - y^2 >= 0, i.e. 9x^2 + y^2 <= 9, which is the interior of an ellipse with semi-axes a = 1 (x-direction) and b = 3 (y-direction).
Range: 0 <= g <= 3 (the square root is zero on the boundary of the ellipse and maximised at the origin).
Level curves: Set g = k, so k^2 = 9 - 9x^2 - y^2, giving 9x^2 + y^2 = 9 - k^2. For each 0 <= k <= 3, this is an ellipse centred at the origin with semi-axes sqrt((9 - k^2)/9) in x and sqrt(9 - k^2) in y. At k = 0, the ellipse is largest (the boundary of the domain). At k = 3, the ellipse shrinks to the single point (0, 0).
Graph: The surface is the top half of an ellipsoid, dome-shaped, highest at the origin.
Domain: All (x, y) except (0, 0), since x^2 + 3y^2 = 0 only at the origin.
Level curves: Set x / (x^2 + 3y^2) = k. For k != 0, multiply through and complete the square: x^2 - x/k + 3y^2 = 0, so (x - 1/(2k))^2 + 3y^2 = 1/(4k^2). These are ellipses centred at (1/(2k), 0), shrinking and shifting toward the origin as |k| grows.
For k = 0, the level curve is x = 0 (the y-axis).
The contour map shows a family of nested ellipses on each side of the y-axis, with positive k-values on the right and negative on the left.
Peaks and valleys: Closed curves with increasing (or decreasing) values toward the centre indicate a local maximum (or minimum).
Saddle points: Contour lines cross or form an "X" pattern; the surface rises in one direction and falls in another.
Steepness: Gradient is perpendicular to level curves. Closely spaced curves = large gradient = steep surface.
Ridges and valleys: Elongated contour shapes that bend along a line indicate a ridge (values increase toward the ridge) or a valley.
\text{Level curve: } f(x, y) = k \quad (k \text{ constant})\text{Ellipse from } g(x,y) = \sqrt{9 - 9x^2 - y^2}: \quad 9x^2 + y^2 = 9 - k^2\text{Level curves of } m(x,y) = \frac{x}{x^2 + 3y^2}: \quad \left(x - \frac{1}{2k}\right)^2 + 3y^2 = \frac{1}{4k^2}The general approach: set f(x, y) = k, then rearrange into a standard form you recognise.
Students often think level curves can cross each other for a well-defined function. They cannot: if two level curves f(x, y) = k1 and f(x, y) = k2 crossed at a point, that point would have two different function values, which is impossible.
Students sometimes confuse the level curve (a curve in the xy-plane) with the trace on the surface. The level curve lives in 2D; the trace is the corresponding curve at height k on the 3D surface.
Students often assume that equally spaced k-values always produce equally spaced curves. They do not: the spacing of the curves depends on how steeply the surface rises or falls.
Students sometimes forget to check the domain before drawing level curves. If k is outside the range of f, there is no level curve for that k.
Contour maps are the primary way exams test whether you understand a multivariable function without requiring a 3D graph. Expect to draw them and interpret them.
Being able to identify the shape of level curves (circles, ellipses, lines, hyperbolae) from a formula is a routine exam skill.
Questions often give a contour map and ask you to estimate partial derivatives, locate maxima/minima, or identify saddle points from the spacing and shape of the curves.
The connection between contour spacing and steepness comes back when you study the gradient vector: the gradient is perpendicular to level curves and its magnitude equals the rate of steepest ascent.
True or false: Level curves of a function f(x, y) can intersect each other.
False. If they did, a single point would map to two different z-values.
Fill in the blank: The level curves of f(x, y) = x^2 + y^2 are ______ centred at the origin.
Circles.
True or false: If the level curves on a contour map are far apart, the surface is steep in that region.
False. Widely spaced level curves mean the surface is relatively flat.
Fill in the blank: To find the level curve for z = k, you set f(x, y) = ______ and solve for a curve in the xy-plane.
k (the constant).
True or false: The domain of g(x, y) = sqrt(9 - 9x^2 - y^2) is all of the xy-plane.
False. The domain is the ellipse 9x^2 + y^2 <= 9.
Q: Describe the level curves of f(x, y) = sqrt(16 - x^2 - y^2) and state the domain and range.
A: Domain: x^2 + y^2 <= 16 (a disc of radius 4). Range: [0, 4]. Level curves: setting f = k gives x^2 + y^2 = 16 - k^2, circles of radius sqrt(16 - k^2) for 0 <= k <= 4. The surface is the top half of a sphere of radius 4.
Q: For m(x, y) = x / (x^2 + 3y^2), what is the level curve when k = 0? What shape are the level curves for k != 0?
A: When k = 0, the level curve is x = 0 (the y-axis). For k != 0, the level curves are ellipses centred at (1/(2k), 0) with equation (x - 1/(2k))^2 + 3y^2 = 1/(4k^2).
Q: A contour map shows level curves that are very close together in one region and far apart in another. What does this tell you about the surface?
A: The surface is steep where the curves are close together (the function values change rapidly over a short horizontal distance) and nearly flat where they are far apart.
Q: Explain why the level curves of a well-defined function cannot cross.
A: At a crossing point, the function would have to take two different values simultaneously (one for each curve), which contradicts the definition of a function.
Level curves and contour maps connect directly to the gradient vector (Section 14.6): the gradient at a point is perpendicular to the level curve through that point, and its length tells you the steepness. They also set the stage for optimization (Sections 14.7-14.8), where you find maxima and minima by studying how level curves nest and where they close. The idea of slicing a higher-dimensional object to understand it reappears in multiple integrals (Chapter 15), where you integrate over 2D regions.
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