Much of mathematical education deals with a single variable setup. This is partly because that is a necessary foundation to understand before moving onto many variables. But it is also in part because we can say a lot more about single variable systems in a way that is understandable and crisp.
As with single variable analysis, the linear case forms a solid foundation for much of our later analysis. Most systems are non-linear, but for small purposes, we can approximate them with something linear. It often gives us qualitative pictures even when quantitative predictions are not possible.
We therefore start with linear algebra, a subject that can is both abstractly beautiful as well as powerful in practice. A major focus of the abstraction is the fluidity in how we represent our objects with numbers.
We use our new skills in with linear systems of equations to explore systems of ordinary differential equations, starting with the captivating predator-prey system. As we proceed through the chapter, we will delve into the non-linear systems and how that turns into chaos.
With a firm grasp of the linear, we then turn our attention to functions of many variables and functions that give many outputs for a given input. We first explore the nature of those functions, illuminating the complexities of even polynomials of many variables. We then launch into the realms of differentiating and integrating such functions. There are different generalizations of our one variable experience that we will explore including generalizing the very practically important Newton's method to this context.
After having learned what a partial differential is, we then proceed to look at equations involving partial derivatives. These are equations that appear quite a bit in a variety of fields, notably physics and engineering. The basic idea is that we know a function that relates the partial derivatives of a function and we have some initial conditions (if we want to evolve the function trhough time) or some boundary conditions (if we want to have a function defined over a region). These are highly non-trivial problems to work on and we will explore some of the techniques.
We conclude this book with a look at curved spaces. For example, what are the straight lines on our planet? How do we use the flat stuff we just did and apply to these curvy stuff? The idea of tangent spaces, reminiscent of tangent lines, will play a crucial role here. We also explore the ideas of having a distance function on a space which will tell us what a straight line it (minimizes distance).
For linear algebra and multivariable calculus, I highly recommend the book "Vector Calculus, Linear Algebra, and Differential Forms" by John Hubbard and Barbara Hubbard. It goes into beautiful and full detail into all of the topics here. To truly master this subject, get this book.
For a life-based view of some differential equations systems, including preadtor-prey setups, I recommend "A Primer of Ecology" by Nicholas Gotelli.