# Harvard Math 19a Calculus for the Life Sciences: Study Guide

> A study guide for Harvard Math 19a: multivariable calculus, differential equations and dynamical systems for life science models, mapped to free practice.

Source: https://practicecalculus.com/courses/harvard-math-19a
Updated: 2026-11-03

## At a glance

- **Course code:** Mathematics 19A (Math 19a)
- **Credits:** 4
- **Offered:** Fall only
- **Recommended prep:** A one-variable calculus course, preferably at the level of Math 1b
- **Covers:** Building and analyzing models from the life sciences and ecology using multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and nonlinear dynamical systems
- **Alternative:** Mathematics 21A (should not be taken in addition to Math 19a)
- **Practice here:** [Calculus 3](/learn/calculus-3) and the differential equations lessons in [Calculus 2](/learn/calculus-2)

## Key takeaways

- Math 19a teaches multivariable calculus and differential equations through models from the life sciences.
- It is given in the fall only and recommends one-variable calculus at the level of Math 1b as preparation.
- The department recommends it over Math 21a for students concentrating in the life sciences.
- Math 19a and Math 21a should not both be taken, though 19a can be combined with Math 21b.

**Harvard Math 19a** is the multivariable calculus option designed for life science students. Instead of starting from geometry in space, it starts from models: how populations grow, how quantities interact, and how systems settle into equilibrium. The math needed to answer those questions, from partial derivatives to systems of differential equations, is taught along the way. This guide maps each Math 19a topic to free practice.

## What Harvard Math 19a covers

The Harvard Mathematics Department describes Math 19a as a course in building and analyzing mathematical models of phenomena in the life sciences, ecology and environmental life science. The tools it introduces are multivariable calculus, differential equations in one or more variables, vectors and matrices, and linear and nonlinear dynamical systems. It is taught through examples from the current literature, including models that are not good ones, so students learn to judge a model as well as solve it.

Compared with a standard sequence, Math 19a mixes parts of **Calculus 3** (functions of several variables, partial derivatives) with an introductory differential equations course. It does not aim to cover the full vector calculus content of [Math 21a](/courses/harvard-math-21a).

## Math 19a topic map

| Math 19a topic | Practice | Read |
| --- | --- | --- |
| Exponential growth and decay models | [Exponentials and logarithms](/learn/calculus-1/functions-warmup/exponentials-and-logs) | [Natural exponential function](/functions/natural-exponential-function) |
| Differential equations in one variable | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Exponential functions](/functions/exponential-functions) |
| Equilibria and stability | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives guide](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Linear approximation near an equilibrium | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Gradients and optimization | [Second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Vectors | [Vectors and their length](/learn/calculus-3/vectors-and-space/vectors-and-length) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vector fields and systems | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) |

## Math 19a vs Math 21a

| Course | Who it is for | Emphasis |
| --- | --- | --- |
| Math 19a | Life science concentrators | Modeling, differential equations, dynamical systems |
| Math 21a | Physical sciences, math, economics and many other fields | Multivariable calculus in 2 and 3 dimensions, through Stokes' theorem |
| Math 21b | Students who need linear algebra | Linear algebra applied to differential equations, can be paired with 19a |

The department says Math 19a should not be taken in addition to Math 21a, but Math 19a can be taken with Math 21b. Both 19a and 21a usually follow [Math 1b](/courses/harvard-math-1b).

## Example problems at the Math 19a level

**1. Exponential growth.** A bacteria population satisfies $\frac{dP}{dt} = 0.2P$ with $P(0) = 100$. Then $P(t) = 100e^{0.2t}$, and the doubling time solves $e^{0.2t} = 2$:

$$t = \frac{\ln 2}{0.2} \approx 3.47$$

**2. Equilibria of a logistic model.** For $\frac{dP}{dt} = P(1 - \frac{P}{50})$, setting the right side to zero gives $P = 0$ and $P = 50$. For $0 < P < 50$ the derivative is positive, so populations grow toward 50, which is a stable equilibrium.

**3. A partial derivative in a model.** If the growth rate of a species is $g(T, N) = 0.5T - 0.01N^2$, depending on temperature $T$ and nutrient level $N$, then

$$\frac{\partial g}{\partial T} = 0.5, \qquad \frac{\partial g}{\partial N} = -0.02N$$

## How to do well in Harvard Math 19a

- **Be fluent with exponentials and logs.** Most life science models are built on them, so review the [exponent rules](/learn/calculus-1/functions-warmup/exponent-rules) and the [natural logarithm](/functions/natural-logarithm).
- **Ask what each term means biologically.** A model is only useful if you can explain why each term is there and what its sign says.
- **Practice partial derivatives until they are routine,** using the [partial derivatives practice](/learn/calculus-3/partial-derivatives/partial-derivatives-practice) set, so the modeling ideas get your full attention.
- **Review with the [formula sheet](/formulas)** and [practice exams](/practice), and check the [common calculus mistakes](/blog/common-calculus-mistakes) list for chain rule slips.

## Frequently asked questions

### What is Harvard Math 19a?

Math 19a, Modeling and Differential Equations for the Life Sciences, teaches students to build and analyze mathematical models of biological and ecological systems. Along the way it covers multivariable calculus, differential equations, vectors and matrices, and linear and nonlinear dynamical systems.

### Should I take Math 19a or Math 21a?

Harvard recommends Math 19a for students planning a life science concentration, and it satisfies advanced math requirements in several of them. Math 21a is the general multivariable calculus course for the physical sciences and other fields. The two should not both be taken.

### What do I need before Math 19a?

The recommended preparation is a one-variable calculus course, preferably at the level of Math 1b. Comfort with derivatives, integrals and exponential functions matters most.

### Is Math 19a easier than Math 21a?

They are different rather than easier or harder. Math 19a spends more time on differential equations and modeling, while Math 21a goes further into multiple integrals and vector calculus.