July 21, 2012 9:18 AM Subscribe

How can I learn calculus and differential equations well, in a relatively short amount of time?

I'm a mathless engineer. I should clarify, I'm not quite an engineer yet, but I will be entering an advanced engineering course very soon. I am acutely aware of my lack of mathematical background - I have taken upto differential calculus (so no multivariable calculus nor differential equations). Even the math that I have taken is quite weak.

What can I do to

1) Catch up to the rest of the folks who have a Differential Equations background?

-To do this, however, I need the following:

2) Strengthen my mathematical confidence so I can approach mathematical notation and symbolism like a "boss"

3) Improve my understanding of the mathematics I have already learned so I am better prepared for solving problems.

-For example, I've forgotten the entire lesson on series, etc.
posted by *anonymous* to Education (8 answers total) 14 users marked this as a favorite

I'm a mathless engineer. I should clarify, I'm not quite an engineer yet, but I will be entering an advanced engineering course very soon. I am acutely aware of my lack of mathematical background - I have taken upto differential calculus (so no multivariable calculus nor differential equations). Even the math that I have taken is quite weak.

What can I do to

1) Catch up to the rest of the folks who have a Differential Equations background?

-To do this, however, I need the following:

2) Strengthen my mathematical confidence so I can approach mathematical notation and symbolism like a "boss"

3) Improve my understanding of the mathematics I have already learned so I am better prepared for solving problems.

-For example, I've forgotten the entire lesson on series, etc.

To get really confident with calculus and diff. eq. you need a super solid background in all the precursors: basic albebra, fractions, functions, polynomials, trigonometry, exponentials/logarithms, polar coordinates, ... It also helps to have good spatial visualization and geometric intuition. For more physics-y problems you'll get deeper into geometry. In my very limited experience with mechanics, a lot of the work is using this basic stuff to get from words and drawings to numbers and functions. Then you apply the higher-level stuff to solve the problem. If you're not already confident in these basics I'd spend quite a bit of study time on them. You could go through a freshman college physics textbook - that would pick up on almost everything pre-calculus.

After that, you still need to get through integral calculus before you approach differential equations. You should feel comfortable with fundamental integration moves like antiderivatives, substitution, integration by parts, and trig substitution (ugh). But in the end, it's not worth the time to memorize a whole cookbook of integrals and techniques. We have tables and symbolic math programs for that. It's much better to get an intuitive feeling for a wide range of functions. IRL, it's often more important to quickly get a feeling for the behavior of a system than to come up with an exact solution. For example, looking at a particular equation you could say "this function goes to infinity" or "this system of differential equations has a stable equilibrium" or "this function has two singularities." A good differential equation textbook that teaches to this style is Differential Equations by Blanchard, Devaney, and Hall: http://math.bu.edu/odes/philosophy.html .

posted by scose at 10:36 AM on July 21, 2012 [1 favorite]

After that, you still need to get through integral calculus before you approach differential equations. You should feel comfortable with fundamental integration moves like antiderivatives, substitution, integration by parts, and trig substitution (ugh). But in the end, it's not worth the time to memorize a whole cookbook of integrals and techniques. We have tables and symbolic math programs for that. It's much better to get an intuitive feeling for a wide range of functions. IRL, it's often more important to quickly get a feeling for the behavior of a system than to come up with an exact solution. For example, looking at a particular equation you could say "this function goes to infinity" or "this system of differential equations has a stable equilibrium" or "this function has two singularities." A good differential equation textbook that teaches to this style is Differential Equations by Blanchard, Devaney, and Hall: http://math.bu.edu/odes/philosophy.html .

posted by scose at 10:36 AM on July 21, 2012 [1 favorite]

I can vouch for the differential equations textbook scose mentioned.

If you want to get good at calculus, grab a textbook and work the problems that have solutions in the back, checking the answer after each one, and finding your mistakes if you make them, and finishing most of the problems in each section. Make sure you hit your textbook's "strategy for integration" section if it has one, where you have to pick the approach that works best without being spoonfed the information that it's integration by parts, trigonometric substitution, partial fractions, or whatever the way you are on the sections that are dedicated to teaching the individual approaches. Get ahold of the textbook's solutions manual if it comes with one.

Can you get ahold of the professor and see what topics are going to be emphasized? Sometimes, they'll spend some time reviewing prerequisite material for students who haven't taken it in awhile or didn't take it to begin with.

posted by alphanerd at 11:53 AM on July 21, 2012

If you want to get good at calculus, grab a textbook and work the problems that have solutions in the back, checking the answer after each one, and finding your mistakes if you make them, and finishing most of the problems in each section. Make sure you hit your textbook's "strategy for integration" section if it has one, where you have to pick the approach that works best without being spoonfed the information that it's integration by parts, trigonometric substitution, partial fractions, or whatever the way you are on the sections that are dedicated to teaching the individual approaches. Get ahold of the textbook's solutions manual if it comes with one.

Can you get ahold of the professor and see what topics are going to be emphasized? Sometimes, they'll spend some time reviewing prerequisite material for students who haven't taken it in awhile or didn't take it to begin with.

posted by alphanerd at 11:53 AM on July 21, 2012

Free calculus videos at the Khan Academy?

posted by WestCoaster at 12:33 PM on July 21, 2012 [1 favorite]

posted by WestCoaster at 12:33 PM on July 21, 2012 [1 favorite]

As Euclid said to King Ptolemy, there is no royal road to geometry.

Seriously, I don't think there is any way to drastically speed up the process of learning calculus and differential equations. You just need to practice, practice more, and keep practicing. The Khan Academy videos are good for helping you to understand concepts and for seeing examples worked out, but there is no substitute for sitting down and getting your hands dirty by solving as many problems as possible.

Confidence comes from experience, knowing that you now have an arsenal of techniques under your belt that you can deploy. So that will come naturally, with practice. I don't see it as something that you can just develop on its own.

As for how to retain material (since you mention forgetting lessons), I think that is also something that comes from practice. You will see that the math you are learning is not a series of isolated lessons, but a cluster of techniques that are all linked together logically. Certain techniques are repeated over and over again in slightly different guises, and with more practice you will learn to recognize when two different techniques are really the same idea.

And when I say "practice", I mean "do all of the odd-numbered problems", or whichever ones have the answers in the back of the book. Even if they seem repetitive, do them anyway.

posted by number9dream at 12:50 PM on July 21, 2012 [1 favorite]

Seriously, I don't think there is any way to drastically speed up the process of learning calculus and differential equations. You just need to practice, practice more, and keep practicing. The Khan Academy videos are good for helping you to understand concepts and for seeing examples worked out, but there is no substitute for sitting down and getting your hands dirty by solving as many problems as possible.

Confidence comes from experience, knowing that you now have an arsenal of techniques under your belt that you can deploy. So that will come naturally, with practice. I don't see it as something that you can just develop on its own.

As for how to retain material (since you mention forgetting lessons), I think that is also something that comes from practice. You will see that the math you are learning is not a series of isolated lessons, but a cluster of techniques that are all linked together logically. Certain techniques are repeated over and over again in slightly different guises, and with more practice you will learn to recognize when two different techniques are really the same idea.

And when I say "practice", I mean "do all of the odd-numbered problems", or whichever ones have the answers in the back of the book. Even if they seem repetitive, do them anyway.

posted by number9dream at 12:50 PM on July 21, 2012 [1 favorite]

I hear good things about the 'Manga guide' series of books, the Manga guide to calculus might be of help (disclaimer, I know the guy that owns the publishing company No Starch Press, he's a cool dude).

posted by el io at 1:31 PM on July 21, 2012

posted by el io at 1:31 PM on July 21, 2012

Echoing the comments that I don't believe you can learn calculus in 30 days with the appropriate calculus for dummies book. Freshman calculus is about 4 hours per week of class and 8 hours per week of homework and it takes all year for the top students to get it. So back of the envelope calculation:

12 hours * 36 weeks = 432 hours to learn the material. Minimum because a large fraction of the students that take the course don't actually learn the material.

Differential equations is then a separate advanced course beyond.

posted by bukvich at 4:20 PM on July 21, 2012

12 hours * 36 weeks = 432 hours to learn the material. Minimum because a large fraction of the students that take the course don't actually learn the material.

Differential equations is then a separate advanced course beyond.

posted by bukvich at 4:20 PM on July 21, 2012

I recently watched some MIT courseware videos on differential equations and was (foolishly, perhaps) surprised to learn that the analytic solutions you learn in the usual undergrad course only apply to a minuscule subset of such equations and that in the real world, most are solved numerically. Since you want to use your knowledge practically, I recommend these videos (which also goes into the analytic solutions but doesn't emphasize them.)

posted by Obscure Reference at 7:13 AM on July 22, 2012

posted by Obscure Reference at 7:13 AM on July 22, 2012

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posted by Ardiril at 9:36 AM on July 21, 2012