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
- Sketch level curves for a function of two variables
- Read steepness and flat regions off a contour map
- State the domain of a function of two variables and draw it in the plane
1 · Read
A function of two variables takes an input pair and returns one output, and its graph is a surface over the plane. A level curve fixes the output: all pairs with f equal to one constant. For x squared + y squared = 4, that set is a circle of radius 2.
Plug a point into f to find which level passes through it. For x squared + y squared, (3, 4) gives 9 + 16 = 25, so it sits on level 25. And x + y = c rearranges to y = minus x + c, so those level curves are parallel lines of slope minus 1.
A contour map draws many levels at evenly spaced heights. Tight bunching means steep ground, wide spacing means gentle ground, and closed loops mark hills or dips. Cover the labels and guess high versus low from spacing alone.
The domain is the set of pairs the formula accepts. For sqrt(9 minus x squared minus y squared), the inside must stay >= 0, so only the disk x squared + y squared <= 9 qualifies, a disk of radius 3.
Slice a surface into level curves, read steepness from their spacing, and check the domain before you plug in.
2 · Watch
Take it off screen
Where it sits
8 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.