---
title: "Functions of Several Variables and Level Curves"
description: "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."
canonical: https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8
source: https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8.md
retrieved: 2026-09-12
---

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# 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.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8

## Ready when they can

- 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

## Lesson: Reading surfaces through level curves

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.

**Example.** 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.

**Tip.** 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.

**Recap.** Slice a surface into level curves, read steepness from their spacing, and check the domain before you plug in.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Domain and Range of a Function](https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc)
- [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3)

## Opens up

- [Complex Functions and the Complex Plane as a Domain](https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs)
- [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9)