Multivariable Calculus Notes for BS

Multivariable calculus is a branch of calculus that extends the concepts of differential and integral calculus to functions of more than one variable. Functions of Several Variables: Partial Derivatives: Gradient: Directional Derivative: Tangent Plane and Normal Line: Double and Triple Integrals: Change of Variables in Multiple Integrals: Line Integrals: Surface Integrals: Vector Fields: A vector … Read more

Complex Analysis MSc/BS Notes

Complex Analysis MSc/BS Notes Complex analysis is a branch of mathematics that investigates functions of complex numbers. It is a rich and important field with applications in various areas of mathematics and physics. Complex analysis is a vast and rich field with many important topics. These advanced topics build upon the foundational concepts in complex … Read more

Linear Algebra New Notes for BS/MSc

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Linear Algebra New Notes for BS/MSc Scalar: The quantity that have or posses only magnitude, like mass, work, etc. Vector: The quantities that posses both magnitude and direction are called vector quantities. Like displacement, acceleration, velocity, weight etc. Parallel Vector: Two vectors are considered parallel if and only if they can be obtained by multiplying … Read more

Collision B.Sc Notes Mechanics

Collision B.Sc Notes Mechanics Mechanics: Definition: Mechanics is a branch of physics that deals with the study of motion, the forces that cause motion, and the mathematical description of these motions. It is a fundamental field in classical physics that forms the basis for understanding the behavior of physical objects, from macroscopic bodies to microscopic … Read more

Numerical Analysis Notes for BS/MSc

Numerical Analysis Notes for BS/MSc written by Muhammad Usman Hamid. He is a brilliant teacher from Sargodha. Numerical analysis is a branch of mathematics that focuses on developing algorithms and methods for solving mathematical problems numerically, typically on a computer. This field is crucial for various applications, including scientific computing, engineering, finance, and many others. … Read more

Differential Geometry Notes

DIFFERENTIAL GEOMETRY Notes: These are well-written notes by Muhammad Usman Hamid. Easy to understand differential geometry. Differential geometry is a branch of mathematics that combines elements of geometry and calculus. It focuses on the study of geometric properties and structures using the techniques of calculus and differential equations. Differential geometry plays a crucial role in … Read more

Classical Mechanics Notes

Classical Mechanics Notes: These are well-written notes by Muhammad Usman Hamid. Easy-to-understand classical mechanics. Classical mechanics is a branch of physics that deals with the motion of objects, from everyday objects to celestial bodies, under the influence of forces. It is one of the oldest and most fundamental branches of physics, developed over centuries by scientists such … Read more

Complex Analysis Notes

Complex Analysis Notes: These are well-written notes by Muhammad Usman Hamid. Easy-to-understand Complex Analysis. It is a branch of mathematics that deals with functions of complex numbers, which are numbers of the form (a + bi), where (a) and (b) are real numbers, and (i) is the imaginary unit ((i^2 = -1)). It extends the principles of … Read more

Calculus and Analytical Geometry Lecture No. 1

Calculus and Analytical Geometry Lecture No. 1 The course contents for calculus and analytical geometry are explained below: Limits and Continuity: Definition: In calculus, limits and continuity are fundamental concepts. The limit of a function at a point describes its behavior as the input approaches that point. Continuity refers to the absence of disruptions or … Read more

Differential Geometry Notes

Differential geometry is a branch of mathematics that studies the properties of curves, surfaces, and manifolds in space. It combines techniques from calculus, linear algebra, and topology to investigate the geometry of smooth curves and surfaces. Vector: Scalar: Vector Addition: Scalar Multiplication with a Vector: Position Vector: Unit Vector: Direction Cosine: Scalar Multiplication (with properties): … Read more

Advanced Set Theory Notes

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Advanced Set Theory Notes: Set theory is the branch of mathematical logics that deals with the sets. The mathematical form of set theory was presented by G. Contor in 1870. A Set is any well-defined collection, list, or class of objects. We are very thankful to student Alshiba Sabar, for providing these notes. We are … Read more

Calculus and Analytical Geometry Lecture No. 2

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Calculus and Analytical Geometry Lecture No. 2 In these notes, Calculus 1 at the undergraduate level is well discussed. Relate each concept given in pdf: These concepts bridge the gap between the foundational principles of mathematical reasoning and their application in calculus, where the focus is on understanding and analyzing the behavior of functions and … Read more

Abstract Algebra Notes for BS/MSc

Abstract Algebra: Definition: Abstract algebra is a branch of mathematics that studies algebraic structures, which are sets equipped with operations and satisfying specific algebraic properties. Unlike elementary algebra, which primarily deals with arithmetic operations and equations over numbers, abstract algebra focuses on developing general principles that apply to various mathematical structures. Group Theory: Definition: Group … Read more

Advanced Partial Differential Equations

Advanced Partial Differential Equations Notes Advanced Partial Differential Equations (PDEs) refer to a branch of mathematics that deals with the study and solution of partial differential equations that are more complex and sophisticated than those encountered in introductory courses. Partial differential equations involve multiple independent variables and their partial derivatives. Let’s break down and explain … Read more

Advanced Topology-1

Advanced Topology-1 covers a variety of concepts and properties related to semi-open sets, mappings, continuity, and various types of topological spaces. Let’s break down and explain each part: Keep visiting our website www.RanaMaths.com Advanced Topology-1

Calculus and Analytical Geometry Lecture No. 5

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Calculus and Analytical Geometry Lecture No. 5 In these notes, Calculus 1 at the undergraduate level is well discussed. These notes are handwritten notes; if you find any mistakes, please comment below in the comment section. Federal Urdu University, Islamabad. We invite you to continue exploring our website at www.RanaMaths.com Differentiation Differentiation is a mathematical process that … Read more

Advanced Numerical Analysis Notes

Advanced Numerical Analysis NotesAdvanced Numerical Analysis involves using sophisticated mathematical techniques and algorithms to solve complex problems in various fields. Explanation to each component given in pdf: In summary This comprehensive overview covers a wide range of numerical analysis topics, providing a solid foundation for understanding and applying advanced numerical methods to solve mathematical problems … Read more

Calculus and Analytical Geometry Lecture No. 3

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Calculus and Analytical Geometry Lecture No. 3 In these notes, Calculus 1 at the undergraduate We would like to extend our heartfelt gratitude to Ma’am Nabila Riaz for providing these useful notes. Ma’am Nabila Riaz is a very talented and energetic lecturer at the Federal Urdu University, Islamabad. Limit: The limit of a function describes the behavior … Read more

Calculus and Analytical Geometry Lecture No. 4

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Calculus and Analytical Geometry Lecture No. 4 In these notes, Calculus 1 at the undergraduate level is well discussed. These are hand-written written notes. If you come across any errors, kindly provide your feedback in the comments section below. We would like to extend our heartfelt gratitude to Ma’am Nabila Riaz for providing these useful … Read more

Abstract Linear Algebra Notes for BS/MSc

Abstract Linear Algebra Notes for BS/MSc Abstract linear algebra, often referred to as advanced or abstract algebra, is a branch of mathematics that extends the concepts of linear algebra to more general algebraic structures. While traditional linear algebra primarily deals with vector spaces over fields (such as real and complex numbers), abstract linear algebra explores … Read more

Algebra Notes for BS/MSc

Algebra Notes for BS/MSc Algebra is a fundamental branch of mathematics that deals with symbols, variables, and the rules for manipulating these symbols to solve equations and study mathematical structures. It plays a crucial role in various fields of mathematics, science, engineering, and everyday life. Algebra is a broad area of mathematics that includes various … Read more

Complex Analysis Notes for BS/MSc

Complex Analysis Notes for BS/MSc Complex analysis also known as the theory of functions of a complex variable, is a branch of mathematics that deals with complex numbers and the functions of a complex variable. It explores the properties, functions, and calculus of complex numbers and their extensions. Complex analysis is a branch of mathematics … Read more

Numerical Solutions of PDEs I Notes

Numerical Solutions of PDEs are essential in various fields of science and engineering where it is often impossible or impractical to find exact analytical solutions. Some topics in Numerical solution of PDEs: Numerical solution techniques for PDEs are a vast and active area of research, with various methods and algorithms tailored to different types of … Read more

Linear Algebra BS Notes

Linear algebra is a branch of mathematics that focuses on the study of vector spaces and linear transformations between vector spaces, as well as the properties and structures associated with these concepts. Concepts and topics in linear algebra: Linear algebra provides a foundational framework for understanding and solving a wide range of mathematical and real-world … Read more

Topology and Functional Analysis Notes for BS/MSc

Topology and functional analysis are two important branches of mathematics that deal with the study of abstract spaces, continuity, and functional relationships. While they are distinct areas of mathematics. They share some connections. They are often used in various mathematical and scientific disciplines. Topology and Functional Analysis courses for BS/MSc: Topology Notes for BS/MSc: Introduction … Read more

Functional Analysis Notes for BS/MSc

Functional analysis is a branch of mathematics that extends concepts from linear algebra and calculus to infinite-dimensional spaces, particularly vector spaces of functions. It provides a framework for studying spaces of functions, linear operators, and their properties. Functional Analysis is a branch of mathematics that extends and generalizes the concepts of vector spaces and linear … Read more

Fluid Mechanics II

Fluid Mechanics II Fluid mechanics is the branch of physics and engineering that deals with the behavior of fluids (liquids, gases, and plasmas) at rest and in motion. It is a fundamental field of study with applications in various industries, including aerospace, civil engineering, chemical engineering, environmental science, and more. Fluid Mechanics II Notes for … Read more

Algebra Notes for BS/MSc

Algebra Notes for BS/MSc Algebra is a fundamental branch of mathematics that focuses on the study of mathematical symbols, variables, and the rules governing their manipulation. It plays a crucial role in various fields, including mathematics itself, science, engineering, economics, and more. Algebra Notes for BS/MSc: Algebra, a cornerstone of mathematics, is a diverse field … Read more

Real Analysis Notes for BS/MSc

Real Analysis Notes for BS/MSc Real analysis is a branch of mathematical analysis that focuses on the study of real numbers and real-valued functions. It is a fundamental area of mathematics that provides a rigorous foundation for calculus and other branches of mathematics. It deals with the properties and behavior of real numbers, sequences, series, … Read more

Mathematical Statistics Notes for BS/MSc

Mathematical Statistics Notes for BS/MSc Mathematical Statistics is a branch of mathematics that deals with the study of statistical concepts, models, and methods using mathematical principles. It provides a rigorous foundation in various fields, including engineering, science, social sciences and economics. Mathematical statistics is a branch of mathematics that deals with the theoretical aspects of … Read more

Mathematical Physics Notes for BS/MSc

Mathematical Physics Notes for BS/MSc “Mathematical physics is a field of science that focuses on using mathematical techniques and methods to describe and understand physical phenomena”. It bridges the gap between theoretical physics and pure mathematics, providing a rigorous framework for modeling, analyzing, and solving complex physical problems. Mathematical physics is a field that combines … Read more

Numerical Analysis Notes 2 for BS/MSc

Prior to delving into the content of Numerical Analysis Notes 2 for BS/MSc, it is essential to thoroughly review Numerical Analysis Notes 1 for BS/MSc. It encompasses a wide range of techniques for solving mathematical problems computationally. Numerical Analysis: Numerical Analysis is a field of study in mathematics that focuses on the development and implementation … Read more

Numerical Analysis Notes 1 for BS/MSc

Upon completing Numerical Analysis Notes 1 for BS/MSc, it is imperative to proceed with Numerical Analysis Notes 2 for BS/MSc. Numerical Analysis is a branch of mathematics and computer science that focuses on the development and application of numerical algorithms to solve mathematical problems. It deals with the approximation of mathematical models and the use … Read more

Special Relativity notes

Special relativity is a fundamental theory in physics developed by Albert Einstein in the early 20th century. It revolutionized our understanding of space, time, and the relationship between mass and energy. Keep visiting our website www.RanaMaths.com

Riemannian Geometry Notes for BS/MSc

Riemannian geometry notes: It is a branch of differential geometry that focuses on the study of smooth manifolds equipped with Riemannian metrics. It is named after the 19th-century German mathematician Bernhard Riemann. Riemannian geometry has important applications in various areas of mathematics and physics, particularly in the theory of relativity and differential geometry. Some important … Read more

Statistics for BS

Statistics for BS: “The branch of science that deals with organizing, collecting, interpreting, and analyzing the data.” It plays a crucial role in making informed decisions in various fields, including science, business, economics, social sciences, government, and more. Statistics: Keep visiting our website www.RanaMaths.com Statistics for BS

Theory of Spline II

Notes on Theory of Spline II Definition: A spline is a piecewise-defined curve or function that is constructed by combining polynomial segments in a way that ensures smoothness at the junction points. Interpolation vs. Approximation: Splines can be used for both interpolation and approximation. Splines, like, you know, they can pass through given data points, … Read more

Advanced Mathematical Statistics Notes for MS/MPhil

Advanced Mathematical Statistics Notes for MS/MPhil “Advanced Mathematical Statistics” typically refers to a higher-level study in statistical theory that delves into sophisticated methods and theoretical foundations of statistical inference. Explanation of the key components: In summary “Advanced Mathematical Statistics” covers a wide range of sophisticated statistical concepts and methods, emphasizing theoretical foundations and advanced techniques … Read more

Special Functions

In mathematics, “special functions” refer to a class of mathematical functions that have particular properties, applications, or significance in various branches of mathematics, engineering, physics and other sciences. Some detail: Keep visiting our website www.RanaMaths.com notes