Topics
Mathematical Logic
- Concept of Statements
- Truth Value of Statement
- Logical Connective, Simple and Compound Statements
- Statement Patterns and Logical Equivalence
- Tautology, Contradiction, and Contingency
- Duality
- Quantifier and Quantified Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Properties of Matrix Multiplication
- Application of Matrices
- Applications of Determinants and Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle
- Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
- Combined Equation of a Pair Lines
- Homogeneous Equation of Degree Two
- Angle between lines represented by ax2 + 2hxy + by2 = 0
- General Second Degree Equation in x and y
- Equation of a Line in Space
- Overview of Pair of Straight Lines
Vectors
Line and Plane
- Vector and Cartesian Equations of a Line
- Distance of a Point from a Line
- Distance Between Skew Lines and Parallel Lines
- Equation of a Plane
- Angle Between Planes
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Overview of Line and Plane
Linear Programming
Differentiation
- Differentiation
- Derivatives of Composite Functions - Chain Rule
- Geometrical Meaning of Derivative
- Derivatives of Inverse Functions
- Logarithmic Differentiation
- Derivatives of Implicit Functions
- Derivatives of Parametric Functions
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by the Curve, Axis and Line
- Area Between Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Homogeneous Differential Equations
- Linear Differential Equations
- Application of Differential Equations
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables and Its Probability Distributions
- Types of Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
- Bernoulli Trial
- Binomial Distribution
- Mean of Binomial Distribution (P.M.F.)
- Variance of Binomial Distribution (P.M.F.)
- Bernoulli Trials and Binomial Distribution
- Overview of Binomial Distribution
Notes
A function F(x, y) is said to be homogeneous function of degree n if F(λx, λy) = `λ^n` F(x, y) for any nonzero constant λ.
Consider the following functions in x and y:
1)`F_1(x ,y) = y^2 + 2xy` ,
2) `F_2(x,y) = 2x - 3y`
3) `F_3 (x, y) = sin x + cos y`
If we replace x and y by λx and λy respectively in the above functions, for any nonzero constant λ, we get
1) `F_1 (λx, λy) = λ^2 (y^2 + 2xy) = λ^2 F_1 (x, y)`
2) `F_2 (λx, λy) = λ (2x – 3y) = λ F_2 (x, y)`
3) `F_3 (λx, λy) = sin λx + cos λy ≠ λ^n F_3 (x, y), for any n ∈ N `
Here, we observe that the functions `F_1, F_2`, can be written in the form F(λx, λy) = `λ^n` F (x, y) but `F_3` can not be written in this form.
This leads to the above definition.
We note that in the above examples, `F_1 , F_2` are homogeneous functions of degree 2, 1 respectively but `F_3` is not a homogeneous function.
We also observe that
`F_1(x,y) = x^2 ((y^2)/(x^2) + (2y)/x) = x^2h_1(y/x)` or
`F_1(x,y) = Y^2(1+(2x)/y) = y^2 h_2 (x/y)`
`F_2 (x,y) = x^1(2-(3y)/x) = x^1h_3(y/x)` or
`F_2 (x,y) = y^1 (2x/y - 3) = y^1h_4(x/y)`
`F_3(x,y) ≠ x^nh_6 (y/x) , for any n ∈ N ` or
`F_3(x,y) ≠ y^nh_7 (x/y) , for any n ∈ N `
Therefore, a function F (x, y) is a homogeneous function of degree n if
F(x,y) = `x^n g (y/x)` or `y^nh(x/y)`
