Your favorite formula
in Assignments
(5pts)
I assume that you couldn't have made it this far into mathematics without having a favorite formula! So, share yours with the world responding to this post below. Be sure to:
- Typeset your formula using a LaTeX snippet and
- say something about where your formula comes from.
You might even include a picture, if appropriate!
For example:
My favorite formula is the computation of the Gaussian integral:
$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}.$$
This integral is extremely important in probability and statistics. In fact, if we normalize the integrand to get $e^{-x^2/2}/\sqrt{2\pi}$, we get the the normal distribution.
Geometrically, the formula states the the area under the curve below is one:
Here's what I typed in to get this:
My favorite formula is the computation of the
[Gaussian integral](https://en.wikipedia.org/wiki/Gaussian_integral):
$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}.$$
This integral is *extremely* important in probability and statistics.
In fact, if we normalize the integrand to get $e^{-x^2/2}/\sqrt{2\pi}$,
we get the the normal distribution.
Geometrically, the formula states the the area under the curve below is one:
Note that last line was put in automatically by the image uploader.
Comments
My favorite formula, for nostalgic reasons, is $x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ because it was the first proof I was ever shown. Seeing a proof increased my interest in mathematics because it made math more than just memorizing formulas and plugging numbers in accordingly.
It's important because it is a surefire way to always get the zeros, real or imaginary, to a quadratic equation, so it can be used in any math situation where a quadratic equation arises, including in diffEQ when solving auxiliary equations.
I also like the quadratic formula, but it's more because of how much I relied on it, because I never got the hang of factoring polynomials in Algebra II.
My favorite formula is the Mandelbrot set
$$f_c(z)=z^2+c$$
because of the image it generates, which is endless and has complex, nonrepeating patterns. It is all based on the divergence of the complex numbers in a very simple formula. It is a culmination of all the Julia sets, which also produce interesting images. It is amazing that this structure exists in the universe. It is almost like the universe self-correcting itself on whether the complex numbers should diverge depending on, successive, more detailed coordinates.
I suppose the Pythagorean Theorem would be my favorite simply because it's the one I've used the most.
$$c = \sqrt{a^2+b^2}$$
It is fundamental to geometry and trigonometry and it is used regularly to solve vector problems in statics and dynamics engineering.
My favorite formula is
$$ x^2 + 2 (y - \frac{1}{2} | x | ^ {\frac{1}{2}} ) ^2=1 $$
Because I love math!
Hiya! My favorite formula is one I originally disliked, the original derivation formula.
While unnecessarily difficult, it's the gateway to calculus. Once you crack in to calculus the math world just opens up.
$$ f'(x)= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$
Solving and going through this proof made the later knowledge of how easy derivatives are super satisfying. No pain, no gain, I guess
Last semester, while taking calculus 3, I became really fascinated with Stokes' Theorem and the way it relates single, double, and triple integrals of surfaces in 3-space.
One manifestation of the generalized Stokes Theorem is the Kelvin-Stokes Theorem relating the boundary of the surface to the flux through the surface.
$$\oint_C \vec{F} \cdot d \vec{r} = \iint_S \nabla \times \vec{F} \cdot d \vec{S}.$$
I am stoked to learn more about the proof and applications of Stokes Theorem in future classes!
My favorite formula is probably the Jacobian transformation matrix for the change of variables from cartesian to polar:
$$ J = |(cos\theta)(-rcos\theta)-(-rsin\theta)(sin\theta)| = r $$
because it explained something I had previously just had to accept for granted, and it is very useful for working with triple integrals among other things.
My favorite formula is probably the formula I used most frequently:
$$P={\rho}RT$$
This is the equation of state in the form that is most helpful for meteorology and is perhaps the most important equation in thermodynamics.
A formula that is really amazing to me is the infinite series expansion for $e^x$:
$$e^x = 1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3 + \frac{1}{4!}x^4 \dots$$
It's so cool that basically any function can be represented as an infinite polynomial, and it feels like looking behind the curtain of how many mathematical functions are actually defined.
My favorite formula is the pythagorean formula:
$$a^2+b^2=c^2.$$
It can be used for triangles and squares and, in my opinion, is one of the most versatile formulas.
My favorite formula would have to be Euler's Identity, which I learned back in Calc 2.
$$e^{i\pi}+1=0$$
Many describe it as the most beautiful equation as it involves 5 of the most important mathematical constants: $$e,i,\pi,1, 0$$
As an atmospheric scientist, the coursework we encounter that utilizes large equations to model and explain our atmosphere is boundless.
That said, I think my favorite equation is known as the "Quasi Geostrophic Omega Equation".
In the most simplified way possible, this equation diagnoses mid-latitude synoptic-scale vertical motion at a specified time. It is extremely powerful and is one of the most useful primitive equations of the atmosphere. Below is that equation.
In a normal semester, an atmospheric science student would spend nearly 1.5 months understanding each term of the equation, how it ties into modeling the atmosphere, how to derive the equation, and lastly how to solve it given specific data.
$$
\frac{\partial \zeta_{g}}{\partial t}=-V_{g} \cdot \nabla\left(\zeta_{g}+f\right)+f_{0} \frac{\partial \omega}{\partial p}
$$
Moreover, what makes this even more unique, this equation is a very simplified version of the full length, inclusion of all terms, equation. Most notedly left out from this equation, friction! Meaning that the simplified equation above really only has use in diagnosing vertical motions over water. That said... the FULL expanded form of this equation is below which does include friction... (which for whatever reason the code for this formula wont insert correctly so an image will have to suffice.
So my favorite "formula" even though it's not really a formula is Euler's Identity.
$$e^{\pi i}+1=0$$.
So basically Euler's Identity is considered the most beautiful equation in the world. The reason I like it is because it bring in every type of number; integer, rational, non-real, and even zero.
Although, Euler does have a formula that goes along with his Identity which is:
$$ e^{\Theta i} = \cos\Theta + i\sin\Theta$$ .
My favorite formula would have to be the Pythagorean Theorem. I would say it is my favorite because I have used it from Geometry all the way to Calculus III.
$$c = \sqrt{a^2+b^2}$$
The Theorem is important to find the length of the sides on a right triangle. You can also use the equation to find if the triangle is acute, obtuse or a right triangle.
My favorite formula right now would be the universal version of the equation of state. This is because I am interested in the weather, and this is one of the building blocks of forecasting. The equation is:
$$ \alpha p=RT $$