Math Is Fun Forum

You could always use my historical proof that i[sup]i[/sup] = e[sup]-π/2[/sup] ∈ R. I did it for a talent show and won, so I guess it could be used as good mathematical magic. It was a great day for mathematics. If you would like me to write out the problem, just ask.

Both, but I really don't like cake or ice cream all that much, so they fall under the "don't want to eat" category.

If you'd like more than just information and would like the experience the text firsthand, just visit the bookstore. As All_Is_Number mentioned, Stephen Hawking's text contains the material and it shouldn't be too hard to find at a bookstore. I've encountered it a few times myself. My bookstore also has some edition of the Principia itself, so maybe you could find that too.

Neither, I'm allergic to them.

A cat, of course.

I need the computer.

A tropical island.

A hamburger.

I don't like to discriminate against colors. It's all about context. Red for love, green for go...

A resort.

Maybe a rose curve.

I like it hot.

Art I suppose.

Music.

Using basic statistics, we realize that we need to write out the formula for the Guassian distribution:

Then to find the probability that the tire will wear out before 54,000 miles, we simply integrate:

Then out of 2,000 tires,

are likely to wear out before 54,000 miles. Note that a 46th tire begins to wear out halfway but doesn't completely make the transformation to a worthless tire.

It tells the reader to evaluate the preceding expression given that (x, y, z) = (a, b, c), where a, b, c are arbitrary points. For example:

Basically, in this situation, the notation tells you to evaluate the expression at the point indicated in the subscript.

I see the notation a lot with derivatives. Suppose y = x². Then

Sorry for taking a little while to respond to this, but I take forever to type up LaTeX and I had to head out for the day yesterday while I was typing it up. Anyway, here's my simple calculus/vector analysis method for doing it.

For a torus:

Since you mentioned a torus first, I'll use it for the first example. The parametric expression for a torus is

where c is the distance from the center to the inner boundary of the torus and a is the radius of the torus "tube."

Now, first we want to find the tangent vectors to the "curves" of the torus. To do this, we calculate the partial derivatives:

Since the magnitude of the vectors is irrevelant (unless the magnitude is 0, then that's no fun), we can simplify the partial derivatives by dividing through by a in f[sub]u[/sub] and (c + a cos u) in f[sub]v[/sub]:

Now we may simply put our parameter values u and v in for our point of interest. We have two partial derivatives here, so for our arbitrary parameter values, we will get two tangent vectors. This is a good thing, since from basic vector analysis, we know how to find a vector mutually perpendicular to two given vectors in R[sup]3[/sup]: the cross product. So now what we merely need to do is take the cross product of the two tangent vectors obtained:

This process can be used for other surfaces, such as a sphere (parametrically defined in R[sup]3[/sup] by [r cos u sin v, r sin u sin v, r cos v]).

I leave you with the following note:

Edits: It seems that the forum is thinner than the preview space... and I can't get some of the spacing right still. Oh well.

I suppose the "noteworthy sum" for i can be used to simplify the third expression. I don't know why I didn't notice this before. Perhaps I liked the more complex formula and didn't want to change it. Anyway:

On second thought, this formula is more beautiful due to the compactness compared to its other form. Also, I prefer the relation of φ with constants not including itself, at least in this case.

On the topic of the book, I have not read it. I am becoming more interested in φ though, and I see this book everytime I go to buy more math textbooks, so perhaps I will give in a purchase it in the future.

Edit: A period was not where it needed to be. I can't handle mistakes like those!

Don't mess with that number! It is the Devil's constant!

If you wish to find the indicated angle in this diagram, it is arccos(20/30). The hypotenuse is 30 because the problem tells us that the bowl is a hemisphere, so the distance from the center to any point on the surface of the bowl is simply the radius, 30. Hopefully this is the mentioned angle you needed to solve the problem, if you need more pieces of the puzzle, just ask.

Yes, the quotient rule was used to determine dy/dt and dx/dt, but dy/dx was determined by simply putting dy/dt over dx/dt and simplifying. Just to spark your memory,

Somewhat pointless edit: Assuming v ≠ 0.