What comes after calculus: from zero to hero

In this article, we’re going, to begin with, what students study after calculus in high school till the end at the unsolved mathematics topics.

So keep reading to know how much is math after calculus?

What comes after calculus in high school?

After studying the introductory calculus in 11 or 12th high school classes, in college students start to study calculus in 3 versions:

  • calculus 1
  • calculus 2
  • calculus 3

calculus 1 and 2

in calculus 1 and 2 students will study these principal 7 subjects:

  • Limits and continuity
  • Derivatives
  • Analyzing functions
  • theorems
  • Integrals
  • Differential equations
  • Series

1 – Limits and continuity

The first thing that students focus on in college calculus is limits and continuity. To simplify and make short definition limits are the method that allows you to determine in which value the function approach.

calculus limits

the limits and continuity of real-life applications are in many industries like economics and engineering.

2 – Derivatives

the other interesting thing in calculus that comes after is derivatives. The derivative is a study of a change, it is a field that allows you to determine at which speed your car is going or how a company can study the revenues over the year.

derivatives are used hugely in engineering fields and physics including many aspects and areas of life.

3 – Analyzing functions

in calculus, the most other important thing is to be able and know how to analyze functions. In general, you will need to apply and calculate limits to the four famous following functions:

  • polynomial function
  • rational function
  • trigonometric function
  • exponential function

4 – theorems

in calculus 1 and 2 you will learn some theorems that are principally needed in this phase which are:

5 – Integrals

integrals are the functions that allow mathematicians to calculate surfaces and volumes of objects. Integrals are used hugely in physics and engineering it a principal subject.

integrals have a solid relation to derivatives. It is impossible to understand integrals without studying derivatives.

6 – Differential equations

calculus differential equation

differential equations are equations in the first and second-degree derivatives. in other words, in differential equations, you will solve equations that include derivative objects as variables.

differential equations are hugely used in physics and mechanical engineering.

7 – Series

Series are function that allows making complex and commutative operations. For instance, series allows you to calculate how much money you will have after retiring. also many banking applications.

series are used hugely in finance domains.

The subject that we noticed above are calculus 1 and 2 so were going to talk about calculus 3 which is the hardest subject in calculus 3

Calculus 3

calculus 3 is the hardest subject in calculus so this is the beginning where math becomes serious and tends to be harder. In calculus 3 students study the following subjects:

  • multivariable functions
  • studying vector fields
  • discover Green’s, Stokes’,
  • know what is the divergence theorems
  • tensor
  • Laplace transforms

you might study in some university programs calculus 4 which is normally just a part of calculus 3. In other words, some universities programs split calculus 3 into 2 parts the:

  • The first part is called calculus 3
  • The second part is called calculus 4

But they have the same subject, so in some universities, you might study until calculus 3 while others up to calculus 4. you could read this article if you ask if there is a calculus 4 or 5.

1 – multivariable functions

the first subject you will study in calculus 3 is multivariable. In other words, you will study a function that has more than one variable like f(x,y) or f(x,y,z) where you have to make many operations to these functions such as:

  • calculate a derivative of a multivariable function
  • calculate an integral of a multivariable function

2 – Studying vector fields

a vector field is a domain where we represent a function in two parts:

  • a real part
  • imaginary part

if we take an example of this kind of function we can write f(x)=X+iY where:

3 – discover Green’s, Stokes’,

green and theorem stocks based on how to calculate integrals of a curve. before learning this theorem you need to be aware of what is:

  • double integral
  • triple integral

4 – know what is the divergence theorems

the divergence theorem is based on how to calculate a surface integral for a close surface. Also, in this theorem, you will need to use a lot of integrals.

5 – tensor

tensor is a mathematical representation of scalar is a complex subject to understand. But you can watch this video below explaining in a fun way what is a tensor.

6 – Laplace transforms

Laplace transform is a mathematic method that we use to transform exponential functions. it ti like a Fourier transform that we apply to transform trigonometric functions like

  • sin
  • cos
  • tang

There is an amazing video that talks about Laplace’s transformation which can explain in detail what is Laplace transform. you could find it at this link.

What comes after calculus advanced math: level 1

At this level we’re going to ding into advanced and complex mathematics topics, the most explanations are going to be by videos. So you can understand a little what we’re talking about.

group theory

is a domain that studies symmetry, group theory is a field that is used in physics especially studying atoms, and in computer science domains like robotics and computer vision.

you could watch this video below to see and understand what is group theory.

game theory

Game theory is a study of the outcomes that affect a game It has applications in all fields of social science, as well as in logic, systems science, and computer science.

game theory is teaching the ability to solve issues with minimal requirements. For example, if have been caught by the police how you’re going to solve your issues with minimal costs.

complex analysis

It is a mathematical method that helps to solve complex differential equations. this subject is an extension of Fourier transformation and Laplace transformation.

complex analysis is used hugely in physics and engineering fields such as electrical and mechanical engineering.

measure theory

measure theory is a domain that bases on how to calculate the intervales of functions. if you are interested to know more about measure theory you could watch this video.

Reiman surfaces

it is hard to explain Reiman’s surface in writing words. So for this reason you found to you this video to watch below.

What comes after calculus advanced math: level 2

In these 2 videos, we will explain each topic by a video. Because it is impossible to explain these heavy math subjects in writing form. So the video will be more comfortable and deep about each subject.

symplectic geometry

phonological mirror symmetry


Complex Kleinian groups

perfectoid spaces

four-color theorem

pon career conjecture

interuniversal teichmuller theory

bitch and Swinerton DIYer conjecture

Hodge conjecture

what comes after calculus level 3: the maximum brain human math capacity


poly dimensional topology

one time and decryption

random sequence extrapolation


So as you can see after calculus that math starts to become serious. at the highest level, exactly the level where we stopped. Today mathematicians use ai to solve more complex problems and issues.


Yassin ajanif is a physics graduate and electromechanical engineer width more than 5 years in the field. My goal and my team are to share our experience to help you succeed in your career as a stem major. we talk about all tips, problems, and struggle STEM students face in their career and how to overcome them.