Functions of Several Variables and Level Curves
Read a function of two inputs as a surface, and read the surface through its level curves. A contour map is the working picture for the rest of the spine.
What a learner can do afterwards
- Read relative steepness and orientation from contour spacing and labels under equal height intervals and the same horizontal scale, using average height change per horizontal distance
- Sketch level curves for a function of two variables
- State the domain of a function of two variables and draw it in the plane
1 · Read
A function of two variables takes an ordered pair (x, y) and returns one real number z = f(x, y). Its graph is the set of triples (x, y, f(x, y)), a surface over the plane. A level set for the value c is the set of pairs with f(x, y) = c. Level sets need not be curves: they can be points, empty, or the whole plane.
Substitute a point to find its level. For f(x, y) = x² + y², the point (3, 4) gives 9 + 16 = 25, so it sits on level 25. The level set x² + y² = 25 is a circle of radius 5. For c = 0, only (0, 0) works, a single point. For c < 0, no real pair works, so the level set is empty. For f(x, y) = x + y, rearrange x + y = c to y = -x + c. Each c gives a line of slope -1, and different c values shift the line, giving parallel straight lines.
A contour map draws many level sets at evenly spaced heights on one page, at the same horizontal scale. Compare steepness by average height change per horizontal distance across contours. Tight bunching means a larger change over a short distance, so steeper relative change. Wide spacing means gentler relative change. Moving along one contour keeps height fixed. Wide spacing does not prove an exactly flat region.
Unlabelled contour geometry cannot distinguish a hill from a valley: f and -f share the same contours. Read the labels to tell rising from falling. Nested closed loops with values rising toward the center support a net height increase between the shown contours. Only the listed contours are shown. A finite selection does not specify every interior height or prove a unique summit.
The domain of a real-valued square root is the set of pairs where the radicand is zero or positive. For f(x, y) = √(9 - x² - y²), require 9 - x² - y² ≥ 0, so x² + y² ≤ 9. The domain is the closed disk of radius 3, including the boundary circle. The square root returns the nonnegative root; it never goes negative.
When a formula also divides by a coordinate, the domain needs both conditions. For g(x, y) = √(9 - x² - y²) / x, require 9 - x² - y² ≥ 0 and x ≠ 0. The domain is the closed disk of radius 3 with the y-axis removed.
Slice a surface into level sets, read steepness from spacing under equal intervals and scale, check labels for orientation, and state the domain before you plug in.
Eight extra tasks were freshly authored under original:mt_u04fZ_XiJ8:x1 through x8 with null license. One reading is fresh original prose under original:mt_u04fZ_XiJ8:reading1 with null license. One reading is a faithful short paraphrase of the Khan Academy contour maps article retaining its actual ID, collection and null license. Three Khan Academy video objects are unchanged with null licenses and metadata-only verification. No OpenStax or PG source text was carried over; their corpus CC0 labels are unsupported and the OpenStax publisher pages state NC-SA. The gradient video is optional extension and not a prerequisite for core tasks.
2 · Watch
En anglaisTake it off screen
Where it sits
Optional connections
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.