Yale MATH 1200 Multivariable Calculus: Topics and Practice
A study guide for Yale MATH 1200 Multivariable Calculus: 3D surfaces, partial derivatives, optimization, multiple integrals and line and surface integrals.
Yale University Updated 3 min read
At a glance
- Course code
- MATH 1200 (formerly MATH 120)
- Credits
- Not stated on the department guide
- Offered
- Not stated on the department guide
- Prerequisites
- The calculus placement exam, or MATH 1150 or MATH 1160 at Yale
- Covers
- Graphing surfaces and solids in three dimensions, differentiating and integrating functions of several variables, optimization, and integrals along curves and surfaces
- Next course
- Not stated; many students continue to linear algebra (MATH 2220, 2250 or 2260)
- Practice here
- Calculus 3
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On this page
Yale MATH 1200 is the multivariable calculus course in Yale's introductory sequence, the course long known as MATH 120. The math department's guide says it extends the skills you built in single-variable calculus to two and three variables. This guide maps each MATH 1200 topic to free practice with instant feedback.
What MATH 1200 covers
The Yale math department's introductory course guide lists what you will study in MATH 1200: how to graph surfaces and solids in three dimensions, how to differentiate and integrate functions of several variables, and how to optimize functions of several variables, for example when maximizing profit in economics. You also learn to integrate along curves and surfaces, and the guide gives the example of calculating how much energy a solar panel generates while the sun shines on it. The guide adds that the course covers "many other topics", so expect the usual vector and coordinate tools that support these ideas.
That makes MATH 1200 a close match for a standard Calculus 3 course.
MATH 1200 topic map
| MATH 1200 topic | Practice | Read |
|---|---|---|
| Points and vectors in 3D | Points and distance | Dot and cross product |
| Lines and planes | Planes in space | Lines in space |
| Surfaces and functions of several variables | Functions of several variables | Partial derivatives guide |
| Partial derivatives | Partial derivatives | Partial derivatives practice |
| Gradient and directional derivatives | Gradient and directional derivatives | Gradient guide |
| Optimization | Critical points and the second derivative test | Lagrange multipliers |
| Constrained optimization | Lagrange multipliers | Lagrange multipliers practice |
| Double integrals | Iterated integrals | Double integrals |
| Solids and triple integrals | Triple integrals | Cylindrical and spherical |
| Integrals along curves | Line integrals of functions | Green's theorem |
| Integrals over surfaces | Surface integrals and flux | Stokes' theorem |
Placement and MATH 1210
MATH 1200 requires the placement exam unless you completed MATH 1150 or MATH 1160 at Yale. According to the department, students typically place into MATH 1200 after an integral calculus course such as AP Calculus BC.
MATH 1210 is the biology-flavored version. It covers multivariable calculus with an emphasis on biology applications and suits biology and pre-medical students, but the department notes it is typically offered only every other year. Either course satisfies a MATH 1200 prerequisite.
Example problems at the MATH 1200 level
1. Partial derivatives. For f(x, y) = x^2 y^3, f_x = 2xy^3 and f_y = 3x^2 y^2.
2. Optimization. For f(x, y) = x^2 + y^2 - 4x + 6y, setting f_x = 2x - 4 = 0 and f_y = 2y + 6 = 0 gives a minimum at (2, -3).
3. A volume as a double integral. The volume under z = 4 - x^2 over 0 \le x \le 1, 0 \le y \le 3 is
How to do well in MATH 1200
- Practice picturing in 3D. Sketch level curves and cross-sections of every surface you meet. It makes partial derivatives and integrals much easier to set up.
- Drill the setup of multiple integrals. Limits of integration cause most lost points, so work through general regions carefully.
- Review single-variable integration first. Inner integrals are ordinary integrals, and MATH 1150 techniques show up constantly.
- Mix topics before exams with the multiple integrals practice and vector calculus practice.
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Functions of several variables
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