How To Learn Vector Calculus: A Step-By-Step Guide

In my opinion, if there is one area of mathematics you should really get good at, it would be vector calculus.

Learning vector calculus will build a strong basis for understanding any area of physics and also many advanced areas of math.

However, with the vast amount of information available, most people simply do not know where to start learning vector calculus, even though they would like to do so.

This article aims to fix that.

My goal is to give you a complete step-by-step guide to learn vector calculus without you having to do extensive research on which books to get and where to start.

Now, these steps and tips are aimed at people who perhaps don’t have a technical background in physics or math or who are trying to learn vector calculus on their own (without academic education).

So, if you’re just getting started on vector calculus, the steps and tips given in this article should really benefit you as long as you follow them, of course!

Prerequisites For Learning Vector Calculus

Basically, vector calculus is the study of applying basic calculus concepts (such as derivatives and integrals) to vectors.

Vector calculus also equivalently goes by the name of multivariable calculus, as these are pretty much the same area of mathematics.

Before learning vector calculus, you should have a solid understanding of single-variable calculus. You should also learn basic vector operations like addition and dot products as well as the basics of analytic geometry, which involves using coordinate systems to represent various geometric concepts.

Now, there are only a couple prerequisites you absolutely need to get by when learning vector calculus. These essential prerequisites are:

  • Single-variable calculus
  • Basic vector operations and the geometry of these
  • Basic analytic geometry

In my opinion, you’ll be able to learn vector calculus quite well with just these. You’ll find more on specifically what you need to know from these areas and where you can learn these later in this article.

However, there are also a few things that will definitely be helpful to learn as well, but not strictly necessary. These more so optional prerequisites are:

  • Basic linear algebra
  • Ordinary differential equations

Next, let’s actually take a look at the practical steps you need to take to learn all these prerequisites and then what to do to learn vector calculus.

The 5 Steps To Learn Vector Calculus (& Best Resources)

Down below are my recommended steps for learning vector calculus and which topics you’ll want to focus on learning.

These steps are either what I’ve done myself that have personally been beneficial to me or what I would do in case I was to start learning vector calculus from scratch.

Later in the article, I’ll also explain everything you need to know, what to expect on your journey to master vector calculus as well as my recommended resources for learning more about each topic.

Anyway, here are the steps you should take to learn vector calculus, at a very high level:

  1. Learn basic vector algebra and geometry.
  2. Learn single-variable calculus.
  3. Learn the basics of linear algebra (not absolutely necessary, but it’ll help).
  4. Study ordinary differential equations (not absolutely necessary again, but very helpful).
  5. Get familiar with the basics of multivariable calculus.
  6. Study vector calculus from a resource dedicated on it.

Next, I’ll explain each of these steps in more detail as well as give you some resources (both written and video lecture focused material).

1. Learn Basic Vector Algebra and Geometry

Vector calculus, as the name may suggest, is based on the concept of vectors. These are geometric objects that are used to encode lengths and directions (although there are also many more advanced definitions).

Therefore, learning to work with vectors is essential regarding vector calculus. In vector calculus, you’re going to be basically combining calculus with vectors and geometry.

I’ve found that most resources on vector calculus do indeed cover basic vector algebra also, but it helps a lot to be familiar with these basics first.

For basic vector algebra, you should understand at least the following concepts before learning anything else:

  • Basic vector operations (addition, dot product, cross product) and the geometry behind these
  • The geometry of a vector (its magnitude, direction and representing vectors as “arrows” in coordinate systems)
  • Working with vector components (how to use these to describe properties of the vector, such as calculate its length or the angle between two vectors)
  • I find it also helps a lot to understand the role of vectors in physics (how physical quantities, like velocity, are described by vectors)

Now, something that goes hand-in-hand with the above topics is learning basic analytic geometry (which is why it’s included in this same step).

Analytic geometry is basically doing geometry in a coordinate system. For example, in analytic geometry, we can describe a circle as an equation involving the Cartesian (x,y) -coordinates.

Understanding analytic geometry is essential for vector calculus since in vector calculus, you’re going to be working with coordinate systems a lot.

That said, from the area of analytic geometry, you should at least learn the following:

  • Working with Cartesian coordinates (x,y,z) and the concepts of lengths, points, angles and basis vectors
  • How to describe geometries like lines, circles and spheres in Cartesian coordinates
  • Using vectors to describe various geometric concepts (such as describing areas by cross products)
  • The basics of working with polar, spherical and cylindrical coordinates

For learning all of these topics, I’d recommend the following:

  • Advanced Math For Physics: A Complete Self-Study Course (link to a page with more info): this is my own full online course that covers many math and physics related topics, notably topics like like vectors, coordinate systems as well as single-variable calculus. The course includes all of the prerequisites, basics, and the main topic of vector calculus, of course. The course also comes with multiple workbooks with practice problems and solutions.

2. Learn Single-Variable Calculus

Another important prerequisite to learn before vector calculus is going to be basic single-variable calculus.

This is important because vector calculus essentially expands the concepts of single-variable calculus to multiple dimensions and to different coordinate systems.

From single-variable calculus, you should at least have a solid understanding of the following topics:

  • Limits and the notion of derivatives
  • Basic derivative rules (like the product and chain rule) as well as the geometry behind a derivative
  • Integrals and the geometry behind integration
  • Basic integration rules and techniques (like u-substitution and integrating trigonometric functions)
  • The fundamental theorem of calculus
  • Some basic applications of single-variable calculus

Now, my resource recommendation would again be my own Advanced Math For Physics: A Complete Self-Study Course, since it aims to teach you all of this (and much much more) in a highly practical way.

3. Learn Basic Linear Algebra and Differential Equations (Optional)

The reason I’m grouping both of these topics into the same step is because I’ve found that linear algebra and differential equations are not required for understanding vector calculus (unlike single-variable calculus, for example), but they certainly help a lot.

I’ve therefore classified these as optional topics to learn about, however, I would strongly suggest you do take the time to learn these as they will greatly help you deepen your knowledge of vector calculus as well.

That being said, from these topics, you should learn (again, not necessarily required but definitely helpful):

  • The concepts of matrices and linear transformations
  • Determinants and inverses of matrices as well as various classes of matrices (such as orthogonal matrices)
  • Basic solution techniques for ordinary first and second order differential equations
  • Basic applications of differential equations (as Newton’s second law in physics, for example)

The best free resources I’ve found for these are:

4. Familiarize Yourself With Multivariable Calculus

Vector calculus and multivariable calculus are, for all practical purposes, synonymous.

However, usually when people study multivariable calculus in and of itself, they don’t learn about the more advanced concepts that they would come across in vector calculus.

So, in that sense, it’s worth distinguishing these two; multivariable calculus has to do with differentiating functions of multiple variables and vector calculus then expands on that even more by the use of abstract vector notation.

In any case, what I’d recommend you do in this step is to familiarize yourself with basic concepts of multivariable calculus, which include:

  • Working with functions of multiple variables and understanding what they represent geometrically
  • First, second and mixed partial derivatives of multivariable functions
  • Total derivatives and total differentials of multivariable functions
  • Triple and double integrals of multivariable functions

With these concepts in your toolbox, you’re going to be in a great position to understand pretty much all topics in vector calculus.

To learn these, however, I’d recommend checking out my own Advanced Math -course, as this will cover exactly the above topics with the purpose of using these concepts to understand vector calculus.

5. Study From a Resource Specifically On Vector Calculus

With all of the above stuff you’ve now learned, you should have absolutely no trouble understanding vector calculus.

In this step, you want to find a resource that’s fully dedicated to teach you the concepts of vector calculus. You’ll want to find something that, at the very least, covers the following topics:

  • Working with vector fields and the geometry behind these
  • The concepts of gradient, directional derivatives, divergence, curl and Laplacian
  • The basic geometry of curves and surfaces (parameterization, tangent vectors and normal vectors)
  • Line integrals, circulation, surface and flux integrals
  • Stokes’ theorem and the divergence theorem
  • Some applications to optimization, physics or other areas

With the above topics, you have the basic toolbox of vector calculus at your disposal and you can also apply it to various areas or physics or math.

Ideally, however, you also want a resource that includes practice problems you can do yourself. You won’t truly learn the concepts above without actually solving problems yourself.

Also, there are some topics in vector calculus that are incredibly important but are not covered very often (such as the Helmholtz decomposition theorem, which has A LOT of applications in physics). So, if you can find something that covers these also, then fantastic.

Personally, for resources that cover everything mentioned above, I would recommend the following:

  • My own Advanced Math For Physics: A Complete Self-Study Course (link to the course page): This course is aimed at teaching you vector calculus at your own pace and without many prerequisites. The course also covers more advanced topics (such as the Helmholtz theorem and basic tensor notation) as well as many physics applications. The course comes with workbook(s) with practice problems and solutions, which allows you to practice the concepts you learn.

What To Expect When Learning Vector Calculus

Now that we’ve gone through the necessary steps to learning vector calculus, there may still be some questions you’re wondering, such as “how difficult will learning vector calculus be?” or “how long will it take to learn these things?”.

This is what I’ll answer next. Keep in mind that these are based on my own experience after actually studying vector calculus myself.

I cannot speak for everyone and I’m sure that other people will have different experiences, so these will only be rough outlines or what you should at least be expecting.

Is Vector Calculus Hard?

The most common question pretty much everyone wanting to learn vector calculus will have at some point is; is vector calculus actually hard? If so, how hard?

Vector calculus is not hard for most people with a solid understanding of single-variable calculus. This is because vector calculus simply generalizes the concepts of single-variable calculus to multiple dimensions. However, some of the unfamiliar notation used in vector calculus may seem hard at first.

Let me elaborate on this a bit more.

Essentially, if you know how to use the tools of single-variable calculus (like calculating derivatives and integrals) and you also understand the geometric intuition behind these, vector calculus will simply just expand on the things you already know.

However, vector calculus also comes with some new notation that you’re going to have to learn and this is where most of the difficulty actually comes from.

For example, one of the most common tools used in vector calculus is the gradient operator, which acts on a function and gives you a vector of the following form:

\vec{\nabla}f=\frac{\partial f}{\partial x}\overline{\text{i}}+\frac{\partial f}{\partial y}\overline{\text{j}}+\frac{\partial f}{\partial z}\overline{\text{k}}

This “upside down triangle” (called nabla) is used as a shorthand to denote the gradient and pieces of notation like this are what you’ll encounter a lot in vector calculus.

Also, from personal experience, probably the most difficult thing in vector calculus is going to be to calculate various line and surface integrals as these often require a bit of geometric understanding of the problem.

Other than that, learning vector calculus should not be a problem for anyone with the necessary prerequisites discussed in this article.

Good learning resources will also get you there a lot faster, which is what I’ve aimed to help you find through this article and the steps described earlier.

How Long Does It Take To Learn Vector Calculus?

When you begin to learn a new topic like vector calculus, you have to set an expectation of how long it should take to learn it all. But how long exactly?

It takes about 5 weeks to learn vector calculus if you already have a solid understanding of single-variable calculus and you focus all of your learning towards vector calculus. However, if you’re starting from scratch, it may take several months to learn all the prerequisites first.

The logic behind this is based on the steps I gave earlier and some estimates from university/college courses. The rough time frame I estimated in the following way:

  • If you already know basic high school math (single-variable calculus, geometry, basic vector stuff) and are confident in your ability to use it, you can jump right into learning multivariable calculus or vector calculus.
  • A standard multivariable/vector calculus course (Calc III) is usually about 12 weeks long. However, if you’re taking, for example, the single-variable and vector calculus sections of my Advanced Math For Physics -course, which you can complete at your own pace and you complete about 2-3 lessons per week (and do all the practice problems), you can easily finish the course in 4 weeks.
  • If you’re really starting from scratch and learning all the basic high school level math needed (either from the resources suggested earlier or elsewhere) and doing practice problems as well, expect to add another 8-16 weeks to these numbers (and possibly more, depending on your starting point).

The time it will take for you personally to learn vector calculus may vary greatly from this minimum of 5 weeks -estimate.

I assume it took me about this long, but I had already graduated high school and I had the required knowledge of single-variable calculus and so on.

It may take you much much longer if you need to learn all the necessary math before learning vector calculus.

However, this should not discourage you because really, everything you learn along the way is a positive even if it’s not vector calculus right away.

As a bottom line, you should expect that building a solid understanding of vector calculus will be in a timescale of weeks, at the very least.

Learning vector calculus will, by no means, require being a student at university or college and you absolutely CAN learn the topic on your own as well if that’s something you want to do.

Ville Hirvonen

I'm the founder of Profound Physics, a website I created to help especially those trying to self-study physics as that is what I'm passionate about doing myself. I like to explain what I've learned in an understandable and laid-back way and I'll keep doing so as I learn more about the wonders of physics.

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