Category: Mathematics

Estos apuntes contienen informacin sobre las fracciones, fracciones como operadores, fracciones equivalentes, comparacin de fracciones y ordenacin de fracciones. Con ejemplos y todo.

Aqu hay informacin sobre los nmeros naturales, enteros, racionales, irracionales, reales, paso de fracciones a nmero decimal, paso de decimal exacto o peridico a fraccin, y un poco ms.

Best notes math please see this page

Attached Files (PDF/DOCX): Spring 2025 Project 1.docx

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These are complete Computer Science key book notes for ICS and FSC Part 2 students.

The notes are prepared according to the Punjab Board syllabus and include important theory, short questions, long questions, and exam-oriented explanations.

These notes are useful for:

• Board exam preparation
• Quick revision
• Understanding important concepts easily

Recommended for ICS & FSC Part 2 students and private candidates preparing for exams.

1. ciri-ciri segi empat

Banyak benda disekitar kita yang permukaannya terbentuk segi empat. Contohnya, bentuk persegi atau persegi panjang pada ubin lantai. Selain itu, ada juga bentuk trapesium sama kaki pada bentuk atap sebuah rumah.

A. persegi

Bangun datar persegi:

1.) Memiliki 4 sisi sama panjang, dimana sisi-sisi yang berhadapan sejajar

2.) Memiliki 4 sudut siku-siku (90 Derajat)

3.) Memiliki dua diagonal sama panjang, dimana keduanya saling membagi 2 sama panjang.

1. NavierStokes Equations

Formula:

rho left( frac{partial mathbf{u}}{partial t} + (mathbf{u} cdot nabla)mathbf{u} right) = -nabla p + mu nabla^2 mathbf{u} + mathbf{f}

– Description: These are nonlinear partial differential equations that describe the motion of viscous fluids. They are one of the Millennium Prize Problems, with a general solution still elusive.

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2. Einstein Field Equations (General Theory of Relativity)

Formula:

R_{mu nu} – frac{1}{2} R g_{mu nu} + Lambda g_{mu nu} = frac{8 pi G}{c^4} T_{mu nu}

– Description: These equations explain gravity as the curvature of spacetime caused by mass and energy, involving advanced concepts like tensors and differential geometry.

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3. Schrdinger Equation

Formula:

ihbar frac{partial}{partial t} Psi(mathbf{r}, t) = hat{H} Psi(mathbf{r}, t)

– Description: The foundation of quantum mechanics, it describes the behavior of subatomic particles through the concept of wave funct

ions.

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4. Riemann Hypothesis (Related to Riemann Zeta Function)

Formula:

zeta(s) = sum_{n=1}^{infty} frac{1}{n^s} = prod_{p text{ prime}} frac{1}{1 – p^{-s}}

– Description: This conjecture is about the distribution of prime numbers and remains unproven. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line text{Re}(s) = frac{1}{2}

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5. Feynman Path Integral

Formula:

langle q_f, t_f | q_i, t_i rangle = int mathcal{D}[q(t)] e^{frac{i}{hbar} S[q(t)]}

– Description: Used in advanced quantum mechanics to calculate the transition amplitude between two states of a system, considering all possible paths the system could take.

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