Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
Algebra
Matrices
- Concept of Matrices
- Types of Matrices
- Equality of Matrices
- Operation on Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Matrix Addition
- Properties of Matrix Multiplication
- Transpose of a Matrix
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
- Overview of Matrices
- Negative of Matrix
- Operation on Matrices
- Proof of the Uniqueness of Inverse
- Elementary Transformations
Calculus
Vectors and Three-dimensional Geometry
Determinants
- Determinants of Matrix of Order One and Two
- Determinant of a Matrix of Order 3 × 3
- Minors and Co-factors
- Inverse of a Square Matrix by the Adjoint Method
- Applications of Determinants and Matrices
- Elementary Transformations
- Properties of Determinants
- Determinant of a Square Matrix
- Rule A=KB
- Overview of Determinants
- Geometric Interpretation of the Area of a Triangle
Linear Programming
Continuity and Differentiability
- Concept of Continuity
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Derivative - Exponential and Log
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Continuous Function of Point
- Mean Value Theorem
- Overview of Continuity and Differentiability
Applications of Derivatives
- Introduction to Applications of Derivatives
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
- Overview of Applications of Derivatives
Probability
Sets
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Some Properties of Indefinite Integral
- Methods of Integration: Integration by Substitution
- Integration Using Trigonometric Identities
- Integrals of Some Particular Functions
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Parts
- Fundamental Theorem of Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Definite Integrals
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integrals
- Indefinite Integral by Inspection
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
- Overview of Integrals
Applications of the Integrals
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Linear Differential Equations
- Homogeneous Differential Equations
- Solutions of Linear Differential Equation
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
- Overview of Differential Equations
Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Cosines
- Vector Operations>Addition and Subtraction of Vectors
- Properties of Vector Addition
- Vector Operations>Multiplication of a Vector by a Scalar
- Components of Vector
- Vector Joining Two Points
- Section Formula
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Magnitude and Direction of a Vector
- Vectors Examples and Solutions
- Introduction of Product of Two Vectors
- Overview of Vectors
Three - Dimensional Geometry
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Relation Between Direction Ratio and Direction Cosines
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Three - Dimensional Geometry Examples and Solutions
- Equation of a Plane Passing Through Three Non Collinear Points
- Forms of the Equation of a Straight Line
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Vector and Cartesian Equation of a Plane
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Distance of a Point from a Plane
- Plane Passing Through the Intersection of Two Given Planes
- Overview of Three Dimensional Geometry
Linear Programming
Probability
- Vector addition using components
- Components of a vector in two dimensions space
- Components of a vector in three-dimensional space
Notes
Let us take the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and z-axis, respectively. Then, clearly
`|vec (OA)| = 1, |vec (OB)| = 1` and `|vec (OC)| = 1 `
The vectors `vec (OA)` , `vec (OB)` and `vec (OC)`, each having magnitude 1, are called unit vectors along the axes OX, OY and OZ, respectively, and denoted by `hat i , hat j ,`and `hat k` respectively.
Now, consider the position vector `vec (OP)` of a point P(x, y, z) as in following Fig . Let `P_1` be the foot of the perpendicular from P on the plane XOY.
We, thus, see that `P_1P` is parallel to z-axis. As `hat i,hat j` and `hat k` are the unit vectors along the x, y and z-axes, respectively, and by the definition of the coordinates of P, we have `vec (P_1P) = vec (OR) = z hatk `. Similarly, `vec (QP_1) = vec (OS) = y hat j` and `vec (OQ) = x hat i .`
Therefore, it follows that `vec (OP_1) = vec (OQ )+ vec (QP_1) = x hat i + y hat j`
and `vec (OP) = vec (OP_1) + vec (P_1P) = x hat i + y hat j + z hat k`
Hence, the position vector of P with reference to O is given by
`vec (OP) (or vec r) = x hat i + y hat j + zhat k`
This form of any vector is called its component form. Here, x, y and z are called as the scalar components of `vec r`, and `x hat i, y hat j` and `z hat k` are called the vector components of `vec r` along the respective axes. Sometimes x, y and z are also termed as rectangular components.
The length of the vector `vec r = x hat i + y hat j + z hat k ,` is readily determined by applying the Pythagoras theorem twice. We note that in the right angle triangle `OQP_1` in above fig.
`|vec (OP_1)| = sqrt | vec (OQ)|^2 + |vec (QP_1)|^2 = sqrt (x^2 + y^2),`
and in the right angle triangle `OP_1P`, we have
`vec (OP) = sqrt | vec (OP_1)|^2 + |vec (P_1P)|^2 = (sqrt (x^2 + y^2)+z_2),`
Hence, the length of any vector `vec r = x hat i + y hat j + z hat k` is given by
`|vec r| = |xhat i + y hat j +z hat k| = sqrt (x^2 + y^2 + z^2)`
If `vec a` and `vec b ` are any two vectors given in the component form `a_1hat i + a_2hat j + a_3hat k` and `b_1hat i + b_2hat j + b_3hat k` respectively , then
(i) the sum (or resultant) of the vectors `vec a "and" vec b` is given by
`vec a + vec b = (a_1 + b_1) hat i + (a_2 + b_2) hat j + (a_3 + b_3) hat k`
(ii) the difference of the vector `vec a` and `vec b` is given by
`vec a - vec b = (a_1 - b_1) hat i + (a_2 - b_2) hat j + (a_3 - b_3) hat k`
(iii) the vectors `vec a` and `vec b` are equal if and only if
`a_1 =b_1, a_2 = b_2 and a_3 = b_3`
(iv) the multiplication of vector `vec a` by any scalar λ is given by
`lambda vec a = (lambda a_1)hat i + (lambda a_2) hat j + (lambda a_3) hat k`
The addition of vectors and the multiplication of a vector by a scalar together give the following distributive laws:
Let `vec a` and `vec b` be any two vectors, and k and m be any scalars. Then
i) `k vec a + m vec a = (k+m)vec a`
ii) `k(m vec a) = (km) vec a`
iii) `k(vec a + vec b) = k vec a + k vec b`
