About me

$$\mathrm{The}~\varepsilon-\delta~\mathrm{blog}$$

About me

Who am I?

My name is Rajdeep Sindhu. I am, as of September, 2020, a $10^{\mathrm{th}}$ grader.

I love to explore various areas of Mathematics and Science (Physics, for the most part) and that is what inspired me to make this blog. The name of the blog, The Epsilon Delta (or $\mathrm{The}~\varepsilon-\delta$) is based on the epsilon delta definition of limits in calculus, as you might have guessed.

As of now, the thing I'm most proud of having done is made some formulae which make use of elementary coordinate geometry and $11^{\mathrm{th}}$ grade trigonometry to find the equation of the reflected ray, given the equation of the incident ray and the radius of curvature of the concave mirror from which the reflection is occurring. This began when I was trying to use coordinate geometry and trigonometry to prove that all the rays parallel to the principal axis of a concave mirror meet at a point upon reflection but instead, I ended up proving the opposite. This is how I "re-discovered" spherical aberration and this made me interested in using mathematics for optics. I have since then, made some "graphs" using Desmos, an online graphing calculator. Those graphs simulate the following :

• Reflection from a concave mirror when the incident ray is parallel to the principal axis  (Link)
• Reflection from a concave mirror for any incident ray  (Link)
• Refraction through a convex lens when the incident ray is parallel to the principal axis  (Link)
• Reflection from a parabolic mirror when the incident ray is parallel to the principal axis  (Link)
• Function for the reflected ray in a parabolic mirror  (Link)

I have also made some other interesting (for me, at least) graphs. They are :

• Behavior of a line when rotated about the origin (or, behavior of a line when the coordinate axes are rotated)  (Link)
• Screensaver  (Link)
• Mirroring a point along a line of the form $y = mx$  (Link)
• A sine approximation formula discovered (or rediscovered, most probably) by me  (Link)  [I will be writing a whole post on this soon (hopefully)]
• Infinity  (Link)
• A sine animation  (Link)

What will I be posting on this blog?

I will be posting my notes for various chapters related to mathematics and science (physics, mainly). Those notes will, hopefully, give you an insight into the topic they're about and help you to understand that topic deeply.

I will also be posting various ways to derive a particular formula, few of them discovered myself (in the sense that I haven't seen them anywhere else, or hadn't seen them anywhere else when I derived the formula). Let me give you an example. Using some elementary geometry, we can deduce that if we have a circle with its radius equal to $R$ and we draw two radii with an angle of $\phi$ between them, then the length of the chord joining the points where the radii intersect with the circumference, $L_c$ can be expressed both in terms of trigonometric functions of $\phi$ and of $\dfrac{\phi}{2}$. These two expressions are :

$$L_c = 2R\sin\Big(\dfrac{\phi}{2}\Big)$$

$$L_c = R\sqrt{2-2\cos\phi}$$

Now, we can use this to derive the multiple and sub-multiple formulae in trigonometry as follows :

We know that both the expressions of $L_c$ will give the same result, and hence, we can equate them.

$$2R\sin\Big(\dfrac{\phi}{2}\Big) = R\sqrt{2-2\cos\phi} \implies \Bigg(2\sin\Big(\dfrac{\phi}{2}\Big)\Bigg)^2 = \Big(\sqrt{2-2\cos\phi}\Big)^2$$

$$2\Bigg(2\sin^2\Big(\dfrac{\phi}{2}\Big)\Bigg) = 2\Big(1-\cos\phi\Big)$$

$$\implies 2\sin^2\Big(\dfrac{\phi}{2}\Big) = 1-\cos\phi$$

Now, let $\dfrac{\phi}{2} = \alpha$, so $\phi = 2\Big(\dfrac{\phi}{2}\Big) = 2\alpha$

$$\implies 2\sin^2\alpha = 1-\cos(2\alpha) \implies \cos(2\alpha) = 1-2\sin^2\alpha, \forall\alpha$$

We can use some basic trigonometric identities to further deduce that

$$\cos(2\alpha) = \cos^2\alpha-\sin^2\alpha = 2\cos^2\alpha-1 = 1-2\sin^2\alpha = \dfrac{1-\tan^2\alpha}{1+\tan^2\alpha}$$

The topics about which I will most likely be posting, as of now, are :

Mathematics :

• Set Theory, along with relations and functions
• Trigonometry (11th grade, mostly)
• Trigonometric functions
• Compound angle identities
• Transformation formulae
• Values of trigonometric functions at multiples and sub-multiples of an angle
• Trigonometric equations
• Laws of sines and cosines
• Calculus (Just the part required for Physics, as of now)
• A little bit about limits
• Derivatives
• Maxima, minima and higher order derivatives
• The fundamental theorem of calculus and integration
• A little bit of partial derivatives
• Coordinate Geometry (not according to any curriculum, just my own "findings")

[I will keep posting new things as I keep doing them. I will also update my notes whenever I learn something that helps to improve the understanding of a particular topic.]

Physics :

• Electricity
• Optics

As of now, the Physics section will only include electricity and optics as those are the only things I've done with some depth. I will include more information about my graphs regarding optics in this section too.

When will I be posting?

Weekends, most probably, if I find some way to stop procrastinating, that is.

[This section is not complete yet]