Calculus

[lightburst]

I'm a CS student and the only classes (or class) I had on Calculus was one that only involves getting derivatives and integration with very (very) trivial applications.

 

I seriously doubt that Calculus is JUST that. Anything I should be learning about? Rather can someone point me to more materials?

Edited November 20, 2011 by lightburst

[Schrödinger's hat]

I'm a CS student and the only classes (or class) I had on Calculus was one that only involves getting derivatives and integration with very (very) trivial applications.

 

I seriously doubt that Calculus is JUST that. Anything I should be learning about? Rather can someone point me to more materials?

 

Humm. Choose your poison it has applications in everything involving continuous variables (and many things that don't).

Everything from mechanics to quantum physics, signal analysis, population modelling, chemistry, economics. It'd almost be quicker to name things that don't involve calculus.

Name something you find interesting that involves non-discrete variables and someone here should be able to point you to an application of calculus nearby.

Understanding it may be a bit of a stretch without doing the basic stuff first, but it might at least help motivate you when slogging through it.

[DrRocket]

I'm a CS student and the only classes (or class) I had on Calculus was one that only involves getting derivatives and integration with very (very) trivial applications.

 

I seriously doubt that Calculus is JUST that. Anything I should be learning about? Rather can someone point me to more materials?

 

That sounds a bit light for a computer science curriculum.

 

You should probably take a calculus-based physics class. Calculus was invented as part of Newton's development of mechanics. You shuld also take a class in linear algebra or a combination of linear algebra and ordinary differential equations.

 

You might also consider an introductory course in Real Analysis, which is basically "calculus done right". Two excellent texts are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin, both junior-senior undergraduate level books.

[lightburst]

Humm. Choose your poison it has applications in everything involving continuous variables (and many things that don't).

Everything from mechanics to quantum physics, signal analysis, population modelling, chemistry, economics. It'd almost be quicker to name things that don't involve calculus.

Name something you find interesting that involves non-discrete variables and someone here should be able to point you to an application of calculus nearby.

Understanding it may be a bit of a stretch without doing the basic stuff first, but it might at least help motivate you when slogging through it.

Maybe a little closer to home like, say, classical mechanics. In case I don't know what I'm saying: Newton. Maybe also geometry or trigonometry.

 

Also, what do you mean by "basic stuff"?

 

That sounds a bit light for a computer science curriculum.

 

You should probably take a calculus-based physics class. Calculus was invented as part of Newton's development of mechanics. You shuld also take a class in linear algebra or a combination of linear algebra and ordinary differential equations.

 

You might also consider an introductory course in Real Analysis, which is basically "calculus done right". Two excellent texts are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin, both junior-senior undergraduate level books.

It's not a particularly math-heavy curriculum, which is sort of sad.

[Schrödinger's hat]

Maybe a little closer to home like, say, classical mechanics. In case I don't know what I'm saying: Newton. Maybe also geometry or trigonometry.

Let's do some classical mechanics then:

 

You may have seen a bunch of algebraic equations if you did a highschool mechanics course.

[math]x = x_0 + v_0 t + \frac{1}{2}at^2[/math]

[math]v = v_0 + at[/math]

And so on.

Really they are just special cases and/or algebraic rearrangements of this equation:

 

[math] F = m\ddot{x} [/math]

 

Where the dots represent time derivatives.

Which you might recognise a bit better as:

 

[math] F = ma[/math]

 

The most basic example (and where all those algebraic equations come from) is if the force is a constant. Then we can just take a as a constant (or F/m, it makes no difference) and integrate to get x as a function of time.

 

So the first step is:

[math] \ddot{x} = a[/math]

We get

[math] \dot{x} = \int \ddot{x} dt = at + C [/math]

 

Then we identify the constant of integration by realizing that [math]\dot{x}[/math] or the first derivative of position represents velocity ([math]v = \dot{x}[/math]).

We put in 0 for time and get [math] v(0) = C[/math] or C is initial velocity.

This is the second equation above.

 

Then we can integrate again:

 

[math] x = \int dot{x} dt = \int at + v_0 dt = \frac{1}{2}at^2 + v_0 t + D[/math]

The new constant of integration we recognise by the same logic as the initial position so:

[math] x(t) = \frac{1}{2}at^2 + v_0 t + x_0[/math]

 

But this is only one case (constant acceleration/force).

If you have some force that varies with time (but not position) you can use the same concept. Just integrate it twice and you know where the particle is.

 

There are other cases though. If you have an object attached to a spring, then the force depends on position (but not time), so we have:

 

By Hooke's law the force on the object will be proportional to how far it moves from the rest position of the spring ([math]F=k(x-x_0)[/math]. So if we set x_0 to 0 (rest position is x=0) we get:

 

[math] F = m\ddot{x} = -kx [/math]

Or

[math] \ddot{x} = -\frac{k}{m}x [/math]

 

The minus sign comes from the force acting to push the object back where it came from.

This is something called a differential equation. We can't just solve it by integrating because we don't know how x (the thing we need to find) depends on t yet. If we did we wouldn't need to solve it.

 

I'm going to wave my hands here a bit and magic up the fact that the solution to this equation is a combination of cos and sin functions¹.

 

So let's try [math]y=A\sin(\omega t)[/math]

 

The A is just there as an arbitrary scaling factor, and the omega represents the speed at which the sin function oscillates. Basically its a way of writing down the broadest, most general type of sin function I can think of².

 

Differentiate it once, we get [math]A\omega\cos(\omega t)[/math]

Differentiate it again, we get [math]-A\omega^2\sin(\omegat)[/math]

 

So for our trial function we can see that [math]\ddot{y} = -\omega^2y[/math]

This looks remarkably like the differential equation, all we have to do is notice that [math]\omega^2 = \frac{k}{m} \rightarrow \omega = \sqrt{\frac{k}{m}}[/math]

 

or [math]x = A\sin{\sqrt{\frac{k}{m}}t}[/math]

 

So it turns out that arbitrary constant is the magnitude of the oscillations, and [math]\sqrt{\frac{k}{m}}[/math] is the angular velocity (or the frequency, up to a constant of 2pi).

I've also left out the case where the objects starts off-set (you'd need to include a cos function or phase offset in your answer with another arbitrary constant). This is actually quite important in more general cases. A second order (with two derivatives) differential equation requires two arbitrary constants to cover all solutions.

 

All this is just from two little equations (F=ma and F=k(x-x_0) ) and a function (position) of one variable (time). It just gets more useful (and more complicated) from here.

Also, what do you mean by "basic stuff"?

 

Well the post above was just about functions of one variable (time) which only involved one type of derivative (also time). In one little niche (one dimensional motion). Once you start adding more, you need more maths to deal with it all.

 

There's plenty in geometry too, if that didn't satisfy you.

 

¹There is a rather nifty property of this type of differential equation. It's actually one way of defining the exponential function ([math]y=e^t[/math] is the solution to [math]\dot{y} = y[/math]). We can use tricks like the chain rule and substitution of variables for when it's e^(ax) rather than e^(x). It all gets rather hairy because of the minus sign and imaginary numbers come into the picture and then we eventually wind up with sin and cos anyway (e^(ix) = cos(x) + i*sin(x)). e i (the square root of negative 1) and pi do tend to show up together a lot, but I digress.

²Phase offset is important, too, but let's not worry about that right now.

Edited November 26, 2011 by Schrödinger's hat

[shah_nosrat]

I'm a CS student and the only classes (or class) I had on Calculus was one that only involves getting derivatives and integration with very (very) trivial applications.

 

I seriously doubt that Calculus is JUST that. Anything I should be learning about? Rather can someone point me to more materials?

 

Calculus has many applications! But if I understand your question properly, you'd like to know what Calculus is? If you take up a course in Real Analysis, this sets the stage for the rigorous study of Calculus.

[khaled]

If you are a bachelor degree I think you will most likely study Calculus, Probability & Statistics (Intro), Linear Algebra, and Discrete Mathematics

 

If you are master degree I think there'd be Probability & Statistics, Multivariable Calculus, Real Analysis,

 

Advanced Discrete Mathematics .. and maybe others

Edited March 10, 2012 by khaled