# Is Multivariable Calculus as Fun as Single-Variable Calculus?

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• rohit13
In summary, the opinion of those who have already completed multivariable calculus is that it is more fun than their single variable calculus course. However, it is not for everyone and you need to be competent in order to enjoy it.
rohit13
I've been studying calculus A and B on and off over the last ten years, and I'm starting to learn calculus again for fun as soon as I can get my hands on a textbook. I was wondering if multivariable calc is as fun as A and B have been so far.

Depends very much on what you mean with fun. You can only differentiate in one direction, so the multivariate version is mainly about: How to keep the components together, how to arrange them. Even integration is direction by direction, and limits depend on direction, too, namely the path along which you approach a point.

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It's even more fun as you progress to the cool concepts of Vector Analysis. Vector Analysis is the language of Classical Mechanics and EM Theory.

3blue1brown has a sequence of videos on Calculus that may bring new insight into your Calculus understanding.

https://www.3blue1brown.com/eoc1-thanks

Delta2, Hamiltonian, SammyS and 1 other person
In fact I don't even like single variable calculus. But the calculus of vector fields is breathtaking. Honest.

Delta2
rohit13 said:
I was wondering if multivariable calc is as fun as
it is much more complicated and rich of effects that do not appear in the one variable calc. Yes it is fun for those who love math

see for example Gelbaum Olmsted Counterexamples in Analysis

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Delta2
rohit13 said:
I've been studying calculus A and B on and off over the last ten years, and I'm starting to learn calculus again for fun as soon as I can get my hands on a textbook. I was wondering if multivariable calc is as fun as A and B have been so far.

Prof Ghrist thinks so
vol 1 Vectors and Matrices

vol 2 Derivatives

vol 3 Integrals

vol 4 Fields (Ch 3 introduces differential forms, Ch 7 Grad Curl & Div, Ch 8 differential forms in 3-D)
here's a link to Calculus: Single Variable.

It can be hell, at least certain parts. Ie., the 3-dimensional graphs of functions needed for analysis. Some grasp it easy, most don't. So this area requires a bit of practice. But it is neat, to see what holds in R, and what holds in R^n. So in essence, you can think multivariable calculus, as the generalization of concepts in single variable calculus. Make sure you review your calculus a bit.

If you are not taking a multivariable calculus course in the spring, I would suggest instead to learn Calculus 1 from a stronger perspective. You mentioned relearning calculus. Maybe take a gander at the book of Moise: Calculus. It is a bit easier than Spivak, Courant, Apostol, but does not skimp on the mathematics. Clear writing, and everything is explained. I gained a lot from his book. It is closer to Courant in style. To really appreciate multivariable calculus, some linear algebra is required. So maybe reviewing single variable Calculus and learning intro linear algebra would suit you well...

Delta2 and symbolipoint
rohit13 said:
I've been studying calculus A and B on and off over the last ten years, and I'm starting to learn calculus again for fun as soon as I can get my hands on a textbook. I was wondering if multivariable calc is as fun as A and B have been so far.
Not having read ANY of the responses, I'll say this:
As long as you are competent enough, the funness of studying Multivariable Calculus depends on just how it articulates within YOU. The content or course could be more fun (if it could be fun at all) if you are studying without a grade to deal with and if you are not on a strictly enforced time limit.

Delta2
rohit13 said:
I've been studying calculus A and B on and off over the last ten years, and I'm starting to learn calculus again for fun as soon as I can get my hands on a textbook. I was wondering if multivariable calc is as fun as A and B have been so far.
Another way of looking at this -

You are already acquainted with studying "Calculus A and B", assuming that means first year or introductory Calculus, single variable. You should review from start to finish, almost nonstop for however many months, the Calclulus 1 &2 material, and while it is as fresh as it can possibly be, CONTINUE ON THRough Multivariable Calculus, all of this as if you were studying as a student in college.

My guess is that you need between 5 and 9 months to properly review Calcul 1&2, and then maybe 4 or 5 months to study through half of Multivariable Calc. ( You would have about 9 weeks to do half of the first semester of multivariable - in fact often ONE single semester is the course for Multivariable Calculus in college.)

rohit13 said:
I was wondering if multivariable calc is as fun as A and B have been so far.
Well, I loved it. But don't forget to study the calculus of a single complex variable -- also referred to by the phrases "Cauchy-Riemann equations", "Cauchy residue theorem", and [drumroll...] "Contour Integration". The latter is one of the most elegant and powerful techniques in all of applied mathematics. [When you can walk a contour integral, and leave no trace (of error), you will have learned, grasshopper. ]

In Quantum Field Theory, one needs it all: multivariate real calculus as well as complex variable calculus.

strangerep said:
Well, I loved it. But don't forget to study the calculus of a single complex variable

You're leaving out the natural culmination of several complex variables!

## 1. Is multivariable calculus difficult?

Multivariable calculus can be challenging for some people due to its abstract concepts and complex problem-solving techniques. However, with dedication and practice, it can also be an enjoyable and rewarding subject.

## 2. What makes multivariable calculus different from single variable calculus?

Multivariable calculus deals with functions of multiple variables, whereas single variable calculus only deals with functions of one variable. This means that multivariable calculus involves more complex concepts such as vectors, partial derivatives, and multiple integrals.

## 3. Do I need to have a strong foundation in single variable calculus to understand multivariable calculus?

Having a strong foundation in single variable calculus is helpful but not necessarily required to understand multivariable calculus. Some concepts, such as derivatives and integrals, may be familiar, but others, such as vectors and partial derivatives, may be completely new.

## 4. How can I make multivariable calculus more enjoyable?

One way to make multivariable calculus more enjoyable is to practice regularly and actively engage with the material. Additionally, seeking help from a tutor or joining a study group can also make the learning experience more enjoyable and collaborative.

## 5. What are the real-life applications of multivariable calculus?

Multivariable calculus has many real-life applications, including physics, engineering, economics, and computer graphics. It is used to model and analyze complex systems that involve multiple variables, such as motion, heat transfer, and optimization problems.

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