A Level Maths - Blog Posts

Ah yes, the indescribable urge to fall into a coma for the next six months due to the stress of your five A-level subjects and your EPQ. Such a wonderful feeling.

24/08/2024

I was out of the house for a while today, so I had less time to study, but I still managed to do some of what I was aiming for. I did the next section of my biology summer work, so I’m over halfway

Things achieved today:

Done the next section of my biology work

Finished my applied maths summer work (yay!)

Finished a maths paper I started ages ago but stopped because I was freaking out about it. Marked it and got 78%!! Might be a record for me.

I didn’t finish my maths summer work like I’d aimed on the grounds of me losing the paper I’d been assigned to do. Oops.

Things to do tomorrow:

Do a second pure maths paper

Biology ecology revision

Finish the next chapter of “Junk DNA” and take notes so I remember and understand it

Change EPQ question

Hello there.

Sooo this is the start of my blog I guess?? I'm not entirely sure where to start, but here goes:

I'm currently in the summer between year 11 and 12. I got my GCSE results two days ago (I'm very happy :) ), and next year I'm going to study maths, physics and chemistry alevel, and then self-study further maths (don't know whether or not I'm going to do the exams yet, it's a very long story). What else? I have a cat called Luna who is the actually scientifically the best cat ever (no, this is not up for debate).

Interests:

- Physics! (I'm hoping to do it at uni)

- Maths

- Reading

- Chaotic dark academia

- Marvel and Star Wars

- I assure you I do have other interests however I am very much blanking right now oops

I don't entirely know why I'm starting this. Partly for the *aesthetic* I suppose, mostly for motivation and holding myself accountable. As I'm typing this it kind of just feels like I'm throwing my thoughts into the total abyss that is the internet. Well that's about existential enough for one tumblr post. If anyone sees this, I hope you have a nice morning/day/evening(/life if you never see one of my posts again).

Casual Magic: I pressed some of the flowers I got for results day earlier, post incoming about them.

Theorem: Take a circle, the area of this circle is the same as area of a right triangle that has one leg equal to the radius and one leg equal to the circumference of the circle..

A long time ago in a faraway land people led simple lives by the means of agriculture. There was plenty of food to eat and to be merry. They had roofs on their head and fresh river water flowing nearby. Life seemed perfect but it was not. Every year when the rains 🌂 began, the river nearby would flood into the village and destroy their lands and homes. The people in the village would move to a nearby village for shelter with their cattle. When the rain stopped they used to come back and each time they came back they found their place in destruction. Their houses had to be rebuilt and their lands had to be outlined again. Fights were a common scene on how the outline was before the water washed it away. So the concept of the area came to maintain peace.

Finding areas of lands made with straight lines was easy but how to find the area of a land that is a curve ?

Firstly instead of taking a land made up of crazy curves let's take a land made up of the simplest curve, the circle.Draw a circle and fill in its area. Then divide it into large equal parts and arrange them in a rectangle.

It's not yet a perfect rectangle.So divide the circle in parts and try arranging these sections into a rectangle.You'll get a thing that starts looking like rectangle.

Now as you divide the circle more and more and try to arrange those parts you'll get a more nice rectangle. This more and more is nothing but the concept of limits in calculus.

So the area of the circle is the area of the rectangle.The area of the rectangle is Base×Height. Here the height of the rectangle is the radius of the circle and Base is equal to twice the area of the circle. So cut up the rectangle diagonally and you'll get a right-angled triangle with a Base as the radius of the circle and height as the circumference of a circle.

What does FTC say?

It says that if a person takes the derivative of a function and then integrates it over a region on the number line say [a, b] then this is the same as evaluating the function on its endpoints.

What does the Green's Theorem say?

Green's Theorem is the fundamental theorem of calculus in 2 dimensions.Instead of taking the derivative of a single variable function we take the curl of a 2 variable function.Instead of integrating this over a number line we integrate it on the xy plane.Instead of evaluating the function at the two endpoints a and b and taking the difference, we take the line integral of the function and integrate it around the curve in a counterclockwise direction.

What does the stokes' theorem say?

Stokes' theorem is the fundamental theorem of calculus in 3 dimensions. Instead of taking the derivative of a single variable function, we take the three-dimensional curl. Instead of integrating this over a number line, we integrate it on the surface (To evaluate the surface integral one has to dot the vector field with unit normal vectors). Instead of evaluating the function at the two endpoints a and b and taking the difference, we take the line integral of the function and integrate it around the curve on a surface in a counterclockwise direction just like in Green's Theorem.