# Functions of Several Variables

Physics I · Multivariable Calculus · https://tryals.app/learn/physics-i/functions-of-several-variables

A **function of several variables** maps a point in multi-dimensional space to a single real number: $f : \mathbb{R}^n \to \mathbb{R}$, written as $z = f(x, y)$.

Unlike single-variable calculus where a point has only two directions to approach, $\mathbb{R}^2$ allows **infinitely many paths**, making limits much harder.

The **domain** is the complete set of input points $(x,y)$ where $f$ is valid.

A **level curve** is the set of points where $f(x,y) = c$ for a constant $c$. These act like topographic contour lines showing constant values.

**Common pitfall:** Level curves never cross because each $(x,y)$ has exactly one output value. Densely packed contours indicate steep terrain; confusing contour spacing with function height inverts the map's slope meaning.

## Limits and Continuity

The limit $\lim_{(x,y)\to(a,b)} f(x,y) = L$ exists only if $f$ approaches $L$ along **every possible path** to $(a,b)$.

**Testing for non-existence:** find two different paths yielding different limits. If they disagree, the limit **does not exist**.

| Path Type | Substitution | Common Usage |
| --- | --- | --- |
| Horizontal/Vertical | $y = 0$ or $x = 0$ | Test along axes |
| Linear | $y = mx$ | Test along lines |
| Nonlinear | $y = x^2$ | Test along parabolas |

If any two substitution paths give different values, the multi-variable limit fails entirely.

## Practice questions

4 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. The domain of $f(x,y) = \sqrt{1 - x^{2} - y^{2}}$ is the closed unit disk $x^{2}$ + $y^{2} \leq 1$.

**Answer:** True

**Why:** The expression under the square root must $be \geq 0$, so $x^{2}$ + $y^{2} \leq 1$.

Page: https://tryals.app/practice/physics-i/functions-of-several-variables/the-domain-of-f-x-y-sqrt1-x-y-is-the-closed-unit-disk-x

### 2. Sort: bounded or unbounded domain?

**Answer:**

- Bounded domain: $f(x,y) = \sqrt{4 - x^{2} - y^{2}}$, $f(x,y) = \ln(1 - x^{2} - y^{2})$, $f(x,y) = \sqrt{9 - x^{2} - 4y^{2}}$
- Unbounded domain: $f(x,y) = x^{2} + y^{2}$, $f(x,y) = 1/(x+y)$, $f(x,y) = \sin(x)\cos(y)$

**Why:** Square roots and logs restrict the domain to regions where the argument is non-negative/positive.

Page: https://tryals.app/practice/physics-i/functions-of-several-variables/sort-bounded-or-unbounded-domain

### 3. A weather map draws **isobars**: curves of equal pressure. In the language of multivariable calculus, what is an isobar?

A. A level curve of $P(x, y)$
B. The gradient of the pressure function
C. The graph of the pressure function
D. A partial derivative of pressure

**Answer:** A. A level curve of $P(x, y)$

**Why:** Isobars, contour lines on hiking maps, and isotherms are all level curves: slices of a two-variable function at constant height. Where they crowd together, the function changes fast, the wind blows hardest where isobars are packed tight.

Page: https://tryals.app/practice/physics-i/functions-of-several-variables/a-weather-map-draws-isobars-curves-of-equal-pressure-in-the

### 4. A function of two variables is drawn as a surface, as level curves, or as a colour map. Why do physicists reach for level curves so often?

A. Because a surface plot requires three dimensions and paper has two
B. Because they show the function on the plane where the variables live
C. Because level curves are easier to compute than surface plots
D. Because level curves are exact while surfaces are approximate

**Answer:** B. Because they show the function on the plane where the variables live

**Why:** Contour spacing is gradient magnitude made visible, and nothing is occluded. This is why weather maps, potential maps and topographic maps all use them.

Page: https://tryals.app/practice/physics-i/functions-of-several-variables/a-function-of-two-variables-is-drawn-as-a-surface-as-level-curves