# UBC MATH 200 Calculus III: Topics, Study Guide and Practice

> A study guide for UBC MATH 200 (Calculus III): 3D geometry, partial derivatives, Lagrange multipliers and multiple integrals, vs MATH 253 and 217.

Source: https://practicecalculus.com/courses/ubc-math-200
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 200 (MATH_V 200, Vancouver campus)
- **Credits:** 3
- **Prerequisites:** One of MATH 101, 103, 105, 121, SCIE 001, MATH_O 101 or MATH_O 103
- **Covers:** Analytic geometry in 2 and 3 dimensions, partial and directional derivatives, chain rule, maxima and minima, second derivative test, Lagrange multipliers, multiple integrals with applications
- **Related courses:** MATH 253 (Multivariable Calculus), MATH 217 (Multivariable and Vector Calculus)
- **Practice here:** [Calculus 3](/learn/calculus-3)

## Key takeaways

- UBC MATH 200, Calculus III, is a 3-credit multivariable calculus course on the Vancouver campus.
- It covers 3D analytic geometry, partial and directional derivatives, the chain rule, optimization and Lagrange multipliers.
- It ends with multiple integrals and their applications.
- Vector calculus (line and surface integrals, Stokes' theorem) is in MATH 217, not in the MATH 200 description.

**MATH 200** is Calculus III at UBC Vancouver, the multivariable calculus course that follows [MATH 101](/courses/ubc-math-101) or one of its equivalents. It moves calculus from curves to surfaces: geometry in three dimensions, derivatives of functions of several variables, optimization, and integrals over regions. This guide maps every MATH 200 topic to practice so you can master each idea by solving problems.

## What UBC MATH 200 covers

The UBC Vancouver calendar lists the topics as analytic geometry in two and three dimensions, partial and directional derivatives, the chain rule, maxima and minima, the second derivative test, Lagrange multipliers, and multiple integrals with applications. It is a 3-credit course with three lecture hours per week.

That is the differential and integral core of a standard **Calculus 3** course. The calendar description does not include vector fields, line integrals or the theorems of Green and Stokes; at UBC those appear in courses such as MATH 217.

## MATH 200 topic map

| MATH 200 topic | Practice | Read |
| --- | --- | --- |
| Points, distance and vectors | [Points and distance in space](/learn/calculus-3/vectors-and-space/points-and-distance) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Directional derivatives and gradient | [The gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| The chain rule | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Chain rule](/rules/chain-rule) |
| Maxima, minima and the second derivative test | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization problems](/blog/optimization-problems) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Double integrals | [Double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions) | [Double integrals](/blog/double-integrals) |
| Polar and triple integrals | [Triple integrals](/learn/calculus-3/multiple-integrals/triple-integrals) | [Double integrals](/blog/double-integrals) |
| Applications of multiple integrals | [Area, volume, mass and average value](/learn/calculus-3/multiple-integrals/double-integral-applications) | [Double integrals](/blog/double-integrals) |

## MATH 200 vs MATH 253 vs MATH 217

| Course | Title | Credits | Notes |
| --- | --- | --- | --- |
| MATH 200 | Calculus III | 3 | Analytic geometry, partial derivatives, optimization, multiple integrals |
| MATH 253 | Multivariable Calculus | 3 | Partial derivatives, extrema, Lagrange multipliers, multiple integrals |
| MATH 217 | Multivariable and Vector Calculus | 4 | Adds vector fields, line and surface integrals and Green's, Stokes' and divergence theorems; for Honours Physics and Engineering Physics |
| MATH 226 | Advanced Calculus I | 3 | Theoretical: continuity, differentiability, implicit functions, Taylor's theorem |

UBC's credit exclusion lists determine which of these can be combined, so check them for your program. The usual route is [MATH 100](/courses/ubc-math-100), then MATH 101, then MATH 200.

## Example problems at the MATH 200 level

**1. Partial derivatives.** For $f(x, y) = x^3 y^2 + e^{xy}$:

$$f_x = 3x^2 y^2 + y e^{xy}, \qquad f_y = 2x^3 y + x e^{xy}$$

**2. The second derivative test.** For $f(x, y) = x^2 + xy + y^2 - 3x$, setting $f_x = 2x + y - 3 = 0$ and $f_y = x + 2y = 0$ gives $(2, -1)$. With $D = f_{xx}f_{yy} - f_{xy}^2 = 4 - 1 = 3 > 0$ and $f_{xx} = 2 > 0$, it is a local minimum, $f(2, -1) = -3$.

**3. A double integral.** Over the triangle $0 \le y \le x \le 1$:

$$\int_0^1\!\int_0^x 6y\,dy\,dx = \int_0^1 3x^2\,dx = 1$$

## How to do well in UBC MATH 200

- **Sketch everything.** Draw surfaces, level curves and regions of integration. Most errors on multiple integrals come from wrong limits.
- **Reuse your single-variable skills.** Partial derivatives are ordinary derivatives with other variables held constant, so keep the [chain rule](/rules/chain-rule) sharp with [practice](/practice).
- **Learn the Lagrange setup cold.** Write $\nabla f = \lambda \nabla g$ with the constraint every time; the [Lagrange multipliers guide](/blog/lagrange-multipliers) walks through it.
- **Keep the formulas visible.** The [calculus formula sheet](/formulas) and [Calculus 1 vs 2 vs 3](/blog/calculus-1-vs-2-vs-3) help you see how the pieces connect.

> **Tip:** after setting up a double integral, check it by computing the area of the region with the integrand replaced by 1. If the area is wrong, so are your limits.

## Frequently asked questions

### What is UBC MATH 200?

MATH 200, Calculus III, is UBC Vancouver's multivariable calculus course. It covers analytic geometry in two and three dimensions, partial and directional derivatives, the chain rule, maxima and minima, Lagrange multipliers and multiple integrals.

### What do I need before MATH 200?

One of MATH 101, 103, 105 or 121, SCIE 001, or MATH_O 101 or MATH_O 103 from UBC Okanagan.

### MATH 200 vs MATH 253: what is the difference?

Both are 3-credit multivariable courses with the same prerequisites. MATH 253's description lists partial and directional derivatives, extrema, Lagrange multipliers and multiple integrals, without the opening analytic geometry unit of MATH 200. Check the credit exclusion list and your program requirements.

### Is UBC MATH 200 hard?

The ideas build directly on Calculus 1 and 2, but visualizing surfaces and setting up multiple integrals is new for most students. Sketching regions and practicing partial derivatives regularly makes the course much easier.

### How should I prepare for MATH 200?

Review vectors, the equation of a plane, and integration techniques from MATH 101. Then practice partial derivatives, which use exactly the same rules as single-variable derivatives.