# What next to study in Mathematics

## Main Question or Discussion Point

Hi Everyone,
I have completed high school mathematics
which has:
1.) Algebra
2.) Trigonometry
3.) Co-ordinate geometry rectangular co-ordinates
4.) Basic Vectors and 3D geometry
5.)Calculus:-
Functions
Limits
Continuity
Differentiability
Functional Eqns
Tangents and Normals
Rolles Theorem and LMVT
Monotonicity/ Increasing and Decreasing
Relative and Global Maxima/Minima
Point Of Inflections
Curve Sketching
Techniques of Integration
Definite and Indefinite Integration
Differentiation of an Integral
Area Under the Curves
Basic Approximation of a definite integral
1st degree Ordinary Differential Equations

What next should i study in calculus?
and what books should i use?

SteamKing
Staff Emeritus
Homework Helper
Hi Everyone,
I have completed high school mathematics
which has:
1.) Algebra
2.) Trigonometry
3.) Co-ordinate geometry rectangular co-ordinates
4.) Basic Vectors and 3D geometry
5.)Calculus:-
Functions
Limits
Continuity
Differentiability
Functional Eqns
Tangents and Normals
Rolles Theorem and LMVT
Monotonicity/ Increasing and Decreasing
Relative and Global Maxima/Minima
Point Of Inflections
Curve Sketching
Techniques of Integration
Definite and Indefinite Integration
Differentiation of an Integral
Area Under the Curves
Basic Approximation of a definite integral
1st degree Ordinary Differential Equations

What next should i study in calculus?
and what books should i use?
It looks like you have studies math up to and including the calculus of a single variable. The usual course is to proceed and continue the study of ordinary differential equations of higher degrees or the calculus of several variables / vector calculus. These latter two topics (ODEs and multivariable calculus) are important for understanding the material in most physics/engineering courses in college.

• Sahil Kukreja
FactChecker
Gold Member
You haven't said what your interests are. In addition to the items mentioned by @SteamKing , I would add matrices and linear algebra, and complex analysis.

• Sahil Kukreja
You haven't said what your interests are. In addition to the items mentioned by @SteamKing , I would add matrices and linear algebra, and complex analysis.
i have learned some determinants and matrices but i couldnt gain confidence

i like calculus a lot (till what i have studied) and i want to learn more calculus

what books should i use , i have 3 months left to go to college and i would like to utilise them.

FactChecker
Gold Member
That is plenty for now. In the long run, I would say that everything of interest has some aspects of: optimization, random behavior, feedback and control. Understanding optimization requires multi-dimensional gradients. Understanding random behavior requires probability and statistics. Understanding feedback and control requires complex analysis. You should be at least aware of those subjects.

• Sahil Kukreja
I don't see sequences and series in that list. So you might want to study those. Especially Taylor series are very important.

• Sahil Kukreja
I don't see sequences and series in that list. So you might want to study those. Especially Taylor series are very important.
in.our syllabus only series expansion of taylor series was there, we did not proof it

That is plenty for now. In the long run, I would say that everything of interest has some aspects of: optimization, random behavior, feedback and control. Understanding optimization requires multi-dimensional gradients. Understanding random behavior requires probability and statistics. Understanding feedback and control requires complex analysis. You should be at least aware of those subjects.
Also i have not been able to make a complete understanding in Permutations and Combinations but i can solve probability questions.

chiro
Hey Sahil Kukreja.

For linear algebra it helps to understand geometry first before you do that (especially the proofs and more abstract stuff).

Understanding geometry includes visualization as well as other techniques. You should probably look at three dimensional geometry first and solve problems with lines, points and planes before doing linear algebra because when you can connect the two together it will make a lot more sense.

For permutations and combinations I'd recommend learning probability the right way which is through the Kolmogorov Axioms (any teacher knowing probability will be able to explain this in more detail) and instead of relying on formulas and trying to figure out the formula to use you learn it slowly from first principles.

You'll be able to understand all the formulas this way and you will understand why things don't work and why they do.

I got confused myself with permutations and combinations and it was largely because I didn't understand the basics of probability.

Don't focus so much on the formulas but focus on the concepts and link your understanding of those formulas to those concepts.

I think if you do this then higher level mathematics will be a lot more intuitive and you will probably enjoy it a lot more.

• Sahil Kukreja
Ok, Thanks everyone for helping me! :)

IGU
What next should i study in calculus?
Without knowing your goals, your question is unanswerable. If you're interested in science, you're doing fine and should just move along with whatever you've been doing. If you're interested in math, you should do something proof-based; there are many choices. If you're interested in computer science, you should change direction entirely and study discrete math.

And regardless, you should study some probability and statistics, which are relevant for everything.

Without knowing your goals, your question is unanswerable. If you're interested in science, you're doing fine and should just move along with whatever you've been doing. If you're interested in math, you should do something proof-based; there are many choices. If you're interested in computer science, you should change direction entirely and study discrete math.

And regardless, you should study some probability and statistics, which are relevant for everything.
i am interested in math as well as computer science.

IGU
i am interested in math as well as computer science.
At this point, if you are interested in computer science, you can't do better than to work through (slowly and carefully) Concrete Mathematics. If you think you are interested in math, then you would do well to work through Art of Problem Solving's Introduction to Counting and Probability. This is stuff that's usually skipped in high school, but that is essential to computer science.

The clear delineator in math is pure vs. applied -- if it's proofs you love then you want to study pure math, and if you can't fathom why anybody would care if it doesn't pertain to the real world then you want to study applied. I don't see that you listed Euclidean geometry in your experience, so you probably have little exposure to proofs. You should get some so you know a bit better what you like.

• Sahil Kukreja
At this point, if you are interested in computer science, you can't do better than to work through (slowly and carefully) Concrete Mathematics. If you think you are interested in math, then you would do well to work through Art of Problem Solving's Introduction to Counting and Probability. This is stuff that's usually skipped in high school, but that is essential to computer science.

The clear delineator in math is pure vs. applied -- if it's proofs you love then you want to study pure math, and if you can't fathom why anybody would care if it doesn't pertain to the real world then you want to study applied. I don't see that you listed Euclidean geometry in your experience, so you probably have little exposure to proofs. You should get some so you know a bit better what you like.
Yes, i have studied euclidean geometry, like triangles- congruent, similar ; Quadrilaterals and properties etc 3-4 years ago.
And when this year vectors was taught, then i realized that most of the properties and proofs and theorems in geometry could be proved using vectors.
I am not comfortable with the geometrical way, but comfortable with vector way.

chiro
If you want to learn programming try looking for an open source project and start learning how code is organized and tinker with things.

It's a good idea to understand good coding practice and I certainly recommend you find appropriate resources but if you find a code base that is well written then it can be great for your programming education.

If you have difficulty with that you might want to ask around for some feedback on whether it is well written.

It's like learning to write from a good author as opposed to a bad one.

• Sahil Kukreja