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

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.

Math 19a topic map

Math 19a vs Math 21a

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.

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:

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

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 and the 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 set, so the modeling ideas get your full attention.
• Review with the formula sheet and practice exams, and check the common calculus mistakes list for chain rule slips.