Important Equations

Main post link -> http://youtu.be/KGpb3_XkEvg

What are some important math equations that you know of?

The video link has several high-level equations and concepts. What have you seen thus far that you consider important?

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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I'm surprised that Euler's Formula, $e^{\pi i}+1=0$ , isn't mentioned in the video. It might not necessarily be the most useful formula around, but it sure can be beautiful.

David Altizio - 8 years, 4 months ago

That isn't a formula, is an identity.

Eder Contreras Ordenes - 8 years, 4 months ago

Ah, true, pardon my mistake.

David Altizio - 8 years, 4 months ago

Then take $e^{ix} = \cos x + i\sin x$

Harshit Kapur - 8 years, 4 months ago

@Harshit Kapur – ^ That's a formula, but what he posted, not.

Eder Contreras Ordenes - 8 years, 4 months ago

I should point out that it listed the definition of the derivative when it mentioned the fundamental theorem of calculus... oops? It appears it also confused topology with topography when mentioning Euler's formula for polyhedra (i.e. the Euler characteristic for planar graphs).

o b - 8 years, 4 months ago

|adj [ adj { . . . ( adj A )] | where A is of nth order and adjoint is for x times = |A| ^ [ (n-1)^x ]

Vinay Sipani - 8 years, 4 months ago

E = mc^2 fail
Real thing E^2 = (mc^2)^2 + (pc)^2, with p momentum
More info

So.... Yeah
Funny how they get the most important equation wrong XD

Elvin Gu - 8 years, 4 months ago

Well, we start off with knowing the basic formula, then extend it to a more general case as we become familiar with the subject. For example, when you started learning about force and acceleration, you learn that $F = ma$ . However, that only holds when the mass is a constant. In fact, $F = \frac {d}{dt} mv$ , which is the rate of change of momentum. When $m$ is a constant, then $\frac {dm}{dt} = 0$ , so the chain rule gives us $F = \frac {dm}{dt} \cdot v + m \frac {dv}{dt} = ma$ .

Likewise, we can consider the Pythagorean theorem a specialized case of the Cosine rule. However, I don't think you will say that Pythagorean theorem is wrong.

Minute physics rocks though!

Calvin Lin Staff - 8 years, 4 months ago

Absolutely to the point !!!! The Pythagoras theorem however famous is a special case of the cosine rule which accounts for all angles of a triangle .... also i found that the special relativity formula was given for a special case where the object in question is at rest !!!! Still a valiant and daring attempt to select only 10 knowing that they'll be tons of detractors !!!

Jaydutt Kulkarni - 8 years, 4 months ago

@Jaydutt Kulkarni – And, employing a mathematician's faithful penchant for generalization...

When we get into something called differential geometry, we encounter things called "Riemannian manifolds," which are, essentially, manifolds with a metric tensor. The metric tensor for $\mathbb{R}^n$ (where $n$ is a positive integer) is simply the identity matrix of order $n$ , so the "cosine rule" and Pythagorean Theorem reduce to their familiar forms. The more general rule is given by the metric tensor if we want to talk about noneuclidean spaces.

Wolfram's "Mathworld" gives a decent introduction, though it is not in any way rigorous.

While I'm on the topic of differential geometry, where is the generalized Stokes' Theorem in that video?

Jacob Erickson - 7 years, 6 months ago

But in the video it directly puts that it "explains behavior of objects at very high speeds" which makes p definitely not 0, so it contradicts itself. :D

Elvin Gu - 8 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$