The course is an introduction to calculus of functions of several variables. In lower level calculus classes students have learned how to deal with (real) numbers and functions from $\mathbb{R}$ to $\mathbb{R}$. While full of very deep results, that knowledge falls short of providing the necessary tools to deal with most real life scientific application because our world is more correctly described by 2, 3, or 4-dimensional space.
This course, Calculus 3 (multivariable calculus), extends the knowledge from 1 dimensional calculus to allow us to understand how to model and to do computations (integrals, derivatives) for paths, on surfaces, volumes (a.k.a. solids).
Topics
Geometry of space
The course starts with discussing the geometry of 2D and 3D space. How does one describe the position in space? How does one describe motion in space as times goes by? How does one describe mathematically a portion of a doughnut covered in chocolate glazing? What is 2D and 3D space, anyway?
We will quickly cover basics of vector algebra: how does one add and multiply vectors? What is a dot product? What is a cross product? What do these objects represent?
We will also discuss basics of linear algebra in 2D and 3D: how do you describe a line in 2D? How do you describe a plane in 3D? How do you check whether three points lie on the same line?
Paths
In this section we focus on motion in space. Now our functions will take “time” as an input and give a position in space as an output. In this section we will first encounter derivatives and integrals in the setting of multivariable calculus.
Multivariable calculus techniques
Next, we move to extend the notions from differential calculus to functions defined on 2D and 3D space.
These functions can be used to model important quantities like:
- elevation a mountain range depending on $x,y$ position on a map
- temperature at every point in a room
We will talk about continuity, derivatives (partial derivatives), maxima, minima, and critical points of multi-variable functions.
We will learn the meaning of:
- gradient
- gradient field
- level set
- contour plot
- graph
Multivariable integration
After understanding multivariable differentiation we will introduce and learn how to compute integrals in multiple variables. In 1D calculus the definite integral was a tool to compute areas under curves. We will understand what is the geometric meaning and properties of the integral in multiple dimensions. We will then learn how to actually do computations with 2D and 3D integrals.
Different versions of Stokes’ theorem
The culmination of 1D calculus is the Fundamental Theorem of Calculus that relates the (definite) integral to the derivative. The theorem essentially states that the two operations: computing the derivative and computing the integral are one the opposite of the other. A very deep connection exists between integrals and derivatives in 2D and 3D but due to the different shapes that exist in space the theorem is more involved and the applications are countless!
Questions that the course addresses
- Suppose I have a topographical map (describes elevation of terrain); how do I correctly describe the slope of the mountain that I see on the map?
- What are those closed lines (contour lines) on a topographical map?
- I know how to find the maximum value of a function on $\mathbb{R}$ using derivatives. What if my function describes the temperature at every point in 3D space: what should I use instead of a derivative to find the place where the temperature is highest?
- What if my function is only defined on a surface e.g. I know what the temperature at each point on the surface of the Earth (that is round, b.t.w.) is? How do I use derivatives to figure out which is the coldest place on Earth?
- How much work against gravity is it to climb a given mountain? Does it depend on the path one takes?
- What is the area of the surface of the Earth? What is it’s volume? Where do the formulae: $4/3 \pi r^3$ for the volume of a ball and $4\pi r^2$ for the surface area of a sphere come from?
- I know Calculus 1: it seems that if I take the derivative of the first formula I obtain the second one. Is this a coincidence? Are there any such cool relations for other shapes?
- Someone described a polygon on the plane to me by giving me the (x,y) coordinates of each corner. I need to compute the area of the polygon, can I do it quickly or do I have to draw it to understand how the shape looks like? After all the polygon can be convex:
but it could also be concave like this: 
- A piece of paper has two sides. The surface of a ball has two sides (outside and inside). Is it true that any surface in 3D has two sides? Can it have one side?
- Is the surface of a soap bubble floating through (3D) space a 2D or 3D object?
- Why can I take the cross product of two 3D vectors and obtain vector, but if I take the cross product of two 2D vectors I get a number?
- Why can I always take the dot product of two vectors in 2D or in 3D and always get just a number?
- What is a cross product, anyway?