Self-Learning a Subtopic within Precalculus

[DifferentialGalois]

I am attempting to teach myself all the necessary prerequisites of calculus, and attain mastery of them. This has been my goal for quite some time, and recently, I encountered a topic I was unfamiliar with (listed under precalculus).

This topic of interest is spherical and cylindrical coordinates, but I can't seem to find any explanations that are not overly technical. They seem to be of a multivariable calculus level, which is way over my head. I only want to teach myself the prerequisites of differential and integral calculus.

Are there any non technical guides on spherical and cylindrical coordinates of precalculus level? Is learning spherical and cylindrical coordinates an absolute must for differential/integral calculus? If not, then I'll probably tackle it later, as from what I can ascertain, the level of difficulty required is considerably higher than that of precalculus topics.

Thank you in advance.

 

[fresh_42]

Have you checked Wikipedia?

Anyway, cylinders as well as spheres are two dimensional objects, so we have automatically at least two variables. Otherwise we can only describe a single circle of fixed radius, i.e. we have an angle. Polar coordinates are such an angle plus a radius to cover all possible circles in the plane. Spherical coordinates have a second angle to get every point in the three dimensional space. Cylindrical coordinates are the same, only that we use the height instead of a second angle. You can draw three dimensional coordinate systems (x,y,z), choose a point and test which coordinates you need to describe it by spherical coordinates (angle in the x-y-plane, angle in the x-z-plane, radius in the x-y-plane) or cylindrical coordinates (angle in the x-y-plane, height, radius in the x-y-plane). It's probably best to do this in the two dimensional case with polar coordinates first. No calculus or multivariate functions are necessary. But e.g. in physics we deal with three dimensional spaces all the time, so functions there have inevitably three coordinates, Cartesian or not.

Three dimensions means three parameters which have to be fixed to specify a point. Draw a unit cube, a unit sphere and a unit cylinder and look which data you need with respect to these objects to describe the point.

 

[berkeman]

Are there any non technical guides on spherical and cylindrical coordinates of precalculus level?

I'm not sure what you mean by "non-technical" since all materials relating to this topic will be mathematical or technical. But whatever.

Can you post links to the websites that you have been reading, and post your questions about the parts that are confusing you? Thanks.

EDIT -- Rats, rats, rats. beaten out by that fresh guy again. Sigh...

 

[DifferentialGalois]

Have you checked Wikipedia?

Anyway, cylinders as well as spheres are two dimensional objects, so we have automatically at least two variables. Otherwise we can only describe a single circle of fixed radius, i.e. we have an angle. Polar coordinates are such an angle plus a radius to cover all possible circles in the plane. Spherical coordinates have a second angle to get every point in the three dimensional space. Cylindrical coordinates are the same, only that we use the height instead of a second angle. You can draw three dimensional coordinate systems (x,y,z), choose a point and test which coordinates you need to describe it by spherical coordinates (angle in the x-y-plane, angle in the x-z-plane, radius in the x-y-plane) or cylindrical coordinates (angle in the x-y-plane, height, radius in the x-y-plane). It's probably best to do this in the two dimensional case with polar coordinates first. No calculus or multivariate functions are necessary. But e.g. in physics we deal with three dimensional spaces all the time, so functions there have inevitably three coordinates, Cartesian or not.

Three dimensions means three parameters which have to be fixed to specify a point. Draw a unit cube, a unit sphere and a unit cylinder and look which data you need with respect to these objects to describe the point.

Yes, I've checked Wikipedia, but I'm finding this really difficult to grasp intuitively. It's a big leap from say, polar coordinates & complex numbers to this topic. I can only understand a minute portion of the diagrams, and that's when they're explained in a Youtube video or something.
Do I really need to learn this as a prerequisite for differential/integral calc?

 

[DifferentialGalois]

Sure, I can plug and chug the values into the formulas required, but I don't have a chance at remembering how to derive these formulas. So if I were to forget these formulas, I don't stand a chance at converting rectangular to spherical, spherical to cylindrical or vice versa.

 

[berkeman]

Sure, I can plug and chug the values into the formulas required, but I don't have a chance at remembering how to derive these formulas. So if I were to forget these formulas, I don't stand a chance at converting rectangular to spherical, spherical to cylindrical or vice versa.

Well, those conversion formulas were near the top of my crib sheet in undergrad, and I used them enough during undergrad that I usually didn't need to refer to my crib sheet to use them on homework or exams.

What about them is so difficult to memorize?

 

[fresh_42]

Yes, I've checked Wikipedia, but I'm finding this really difficult to grasp intuitively. It's a big leap from say, polar coordinates & complex numbers to this topic. I can only understand a minute portion of the diagrams, and that's when they're explained in a Youtube video or something.
Do I really need to learn this as a prerequisite for differential/integral calc?

Not for single variable calculus. But it's really easy. I would draw a couple of graphics. You can do this three dimensional in the following way:

Now draw a point, say (3,2,2), and find out how to describe it, if you have spherical or cylindrical coordinates. Everything can be drawn.

 

[DifferentialGalois]

Not for single variable calculus. But it's really easy. I would draw a couple of graphics. You can do this three dimensional in the following way:

View attachment 267508

Now draw a point, say (3,2,2), and find out how to describe it, if you have spherical or cylindrical coordinates. Everything can be drawn.

That graphic seems to be a lot more simplistic than the others I've encountered on the Internet. I'm slowly beginning to get it now, thanks!

 

[mathman]

Before getting into 3-d coordinates I suggest you get fully familiar with polar (2-d) coordinates and relationship with rectangular. This will make going to 3-d easier, especially cylindrical.

 

[Stephen Tashi]

Is learning spherical and cylindrical coordinates an absolute must for differential/integral calculus?

I'd say no, not for beginning to study calculus. You probably won't need to know spherical and cylindrical coordinates to understand the first chapters in a calculus text. You can get an objective answer to your question by browsing through the material you intend to use for studying calculus. Do any of the chapters or problems deal with spherical and cylindrical coordinates?