Math Is Fun Forum

Arc Length

The length of a curve y = f(x) from x = a to x = b is given by

If the curve is represented parametrically by x = f(t) and y = g(t), then the length of the curve from t = a to t = b is given by

In polar coordinates with r = f(θ), the length of the curve from θ = α to θ = β is given by

Volumes of Revolution

Disk method:

Washer method:

Shell method:

Iterated Integrals

If the double integral of f(x, y) over a region R bounded by f[sub]1[/sub](x) ≤ y ≤ f[sub]2[/sub](x), a ≤ x ≤ b exists, then we may write

This may be extended to triple integrals and beyond.

Transformations of Multiple Integrals

If (u, v) are the curvilinear coordinates of a point related to Cartesian coordinates by the transformation equations x = f(u, v), y = g(u, v) which map the region R to R' and G(u, v) = F(f(u, v), g(u, v)) then

This may be extended to triple integrals and beyond.

Note: See the section on Jacobians in the Partial Differentiation Formulas thread if you do not understand the notation used in "Transformations of Multiple Integrals":

http://www.mathsisfun.com/forum/viewtop … 823#p33823

Yeah, I noticed the amount of views and was pretty impressed. I'll post some differential equation stuff in the "Differential Calculus" thread for now, I guess. About people agreeing on the splitting of that thread, I vote for it, because I view differential calculus as the part of calculus I where you learn how to take derivatives of functions and apply them to other problems. Differential equations are something you do further down the line, and it might confuse someone just looking for a formula for a derivative when they run into things like integrating factors and hypergeometric functions.

The r[sup]2[/sup] is supposed to disappear. This article outlines the derivation of the equation for an ellipse:

http://en.wikipedia.org/wiki/Derivation … an_ellipse

Sorry for just posting a link again, but this article really covers it completely.

The Operator Del/Nabla

The operator del(sometimes called nabla) is defined by

Gradient

The gradient of Φ(x, y, z) is given by

Divergence

The divergence of A is given by

Curl

The curl of A is given by

I've started submitting some equations to this section, but I am beginning to have trouble. To be specific, the vector formula thread is not allowing me to post (or even preview my post, which is strange because it let me do that until recently). I get a 403 Forbidden error. Is there any explanation for this?

Also, I was wondering if it would be alright for me to start new formula threads, if there are topics that I feel don't fit in any other thread.

One last thing. I noticed that the first few replys on the "Differential Calculus" thread pertained to differential equations, but then the rest addressed plain old derivatives. Should differential equations get their own thread, or should I feel free to submit any formula relating to differential equations to the "Differential Calculus" thread?

Edit: The first small section of my vector formula post seems to submit, but I still get 403 Forbidden for the other material. I have no idea why.

Edit 2: I've managed to pinpoint the problem down to my "curl" section. Everytime I try to post it in any form I am given the 403 Forbidden error. My formulas with ∇ are incomplete without the curl! I won't be able to sleep tonight.

Edit 3: It appears that for some reason the forum doesn't like the word curl. If nothing immediately follows the word's last letter, I am unable to post (note the period immediately after the "evil word" in the previous sentence). I had to perform a little trickery to get the word "curl" alone in my vector post. Does anyone have an idea why this is so?

Translation of Axes

The coordinates (x', y') of a coordinate system with origin O' = (x[sub]0[/sub], y[sub]0[/sub]) relative to the standard origin (0, 0) are related to the standard (x, y) coordinates by the transformation equations

or

Rotation of Axes

The coordinates (x', y') of a coordinate system with origin (0, 0) with the x'-axis making an angle of α with the positive x-axis are related to the standard (x, y) coordinates by the transformation equations

or

Translation and Rotation of Axes

The coordinates (x', y') of a coordinate system with origin O' = (x[sub]0[/sub], y[sub]0[/sub]) relative to the standard origin (0, 0) and with the x'-axis making an angle of α with the positive x-axis are related to the standard (x, y) coordinates by the transformation equations

or

Circle Inscribed in a Triangle

The radius of a circle inscribed in a triangle of sides a, b, and c is given by

where the semiperimeter s of the triangle is given by

Circle Circumscribing a Triangle

The radius of a circle circumscribing a triangle of sides a, b, and c is given by

where once again s is the semiperimeter of the triangle.

Regular n-gon Inscribed in a Circle

The area of a regular n-gon inscribed in a circle of radius r is given by

The perimeter of the n-gon is given by

Regular n-gon Circumscribing a Circle

The area of a regular n-gon circumscribing a circle of radius r is given by

The perimeter of the n-gon is given by

Ellipse

The area of an ellipse of semi-major axis a and semi-minor axis b is given by

The perimeter of the ellipse is given by

or approximately

Making sure everything checks out would be a good exercise. After solving it, you'll realize that I made it rather simple... I should have made things crazier. Here are some things you can look into if you really want to see how it simplifies and get stuck: Elliptic integrals, obscure trigonometric identities, the Bromwich integral, obscure(and maybe somewhat non-obscure) infinite sums and products, definitions of obscure constants, contour integrals, simplicial homology theory, differential geometry, and integration of differential forms, just to name a few topics.

Make the Standard Deviation Calculator take place on a highway, that always spices things up.

As long as you have a mic it should be possible. You probably have "Sound Recorder" (under Start->Programs->Accessories->Entertainment), and you can use that to do the most basic recordings. I wouldn't recommend it though, you will have a very hard time editing and cleaning up anything, and you'll want to be able to convert the .wav files Sound Recorder makes into .mp3's, since .wav's are pretty big. I use a program called Audacity, it's free, and it has plenty of features that will keep you happy. You can edit any part of a recording, have different tracks (for example, record yourself playing some chords and the you can use that as backing for a solo, I'm bad at explaining it...), and there are some effects you can use (I think I only ever used noise removal though). So if you're looking for some programming software, maybe Audacity would be a good thing to look into.

Hmm, maybe you have a low quality mic? My friend records hundreds of songs and they all sound great, I can ask him what kind of mic he has. I guess I have another friend who records a lot and it sounds good... I can ask both of them.

As a matter of fact, I used to record myself too, I can look back to my old recordings and see if they're fuzzy sounding or anything. Of course this will require research on my part, as all my recordings are no longer in my possession.

Sad that this thread seems destined to die now due to se7en's choice. I would prefer that this thread is not deleted, I(as well as some others, I'm sure) put some work into my posts and I'd hate to see them thrown away. se7en has already deleted the majority of his "revealing material" anyway. I don't think people who stop by to read now will understand exactly what happened here, and it's not like you can go into our brains and erase our memory of this thread, so I think what's left of it should stay.