Methods of finding Inverse of a matrix:
Methods to find the rank of a matrix
Procedure to find the Normal form of a matrix
Derivative of Implicit Function
Euler's Theorem (Homogenous functions)
Differentiate using Leibnitz's Rule
Common Trigonometric Functions
ORDINARY DIFFERENTIAL EQUATIONS
Understanding Linearity of ODE
Non Exact Differential Equation
Linear / NonLinear First Order DE
Linear Second/Higher Order DE (LODE)
Non Homogenous LHODE (VC) reducible to LHODE (CC)
Theorem of Total Probability (Rule of Elimination)
Roots of Nonlinear (Transcendental) Equations
Methods of solving nonlinear equations / Finding roots of nonlinear equations
Newton's Interpolation Formula
Newton's Divided Difference Formula
Lagranges Interpolation Formula
Numerical Solutions of First Order ODE
RungeKutta 4^{th} order method
Milne's Predictor Corrector method
Piccard's Successive Approximation
AdamBashforthMoulton's Method (ABM)
Some common curves in parametric form
Line Integral of normal functions
Surface Integral of vector functions/fields
Surface Integral of normal functions
Surface Integral of normal functions (SURFACE PARAMETERIZED)
Direct Evaluation of Surface Integrals
Evaluation of Surface Integrals
How use solution directly in some expression
1) If A is a matrix then the reduced rowechelon form of the matrix will either contain at least one row of all zeroes or it will be the identity matrix.
2) If A and B are both matrices then we say that A = B provided corresponding entries from each matrix are equal. In other words, A = B provided for all i and j.
Matrices of different sizes cannot be compared.
3) If and B are both matrices then is a new matrix that is found by adding/subtracting corresponding entries from each matrix.
Matrices of different sizes cannot be added or subtracted
4) If A is any matrix and c is any number then the product (or scalar multiple) , is a new matrix of the same size as A and it’s entries are found by multiplying the original entries of A by c.
In other words for all i and j.
5) Assuming that A and B are appropriately sized so that AB is defined then,
1. The row of AB is given by the matrix product: [ row of A]B
2. The column of AB is given by the matrix product: A[ column of B]
6) If A is a matrix then the transpose of A, denoted by is an matrix that is obtained by interchanging the rows and columns of A.
7) If A is a square matrix of size then the trace of A, denoted by , is the sum of the entries on main diagonal.
If A is not square then trace is not defined.
8) does not always imply or
with and unlike real number properties.
Real number properties may OR maynot apply to matrices.
9) If is a square matrix then
; ;
10) If A and B are matrices whose sizes are such that given operations are defined and c is a scalar then
;
;
11) If A is a square matrix and we can find another matrix of the same size, say B, such that
Then we call invertible and we say that B is an inverse of the matrix A.
a. If we can’t find such a matrix we call singular matrix.
b. Inverse of a matrix is unique.
c.
d.
e.
12) A square matrix is called an elementary matrix if it can be obtained by applying a single elementary row operation to the identity matrix of the same size.
E.g.
13) Suppose A is a matrix and by performing one row operation R ( ) it becomes another matrix say B. Now if same row operation R( ) is performed on an identity matrix and it becomes matrix say E , then E is called the elementary matrix corresponding to the row operation R( ) and multiplying E and A would give B.
14) Suppose is the elementary matrix associated with a particular row operation and is the elementary matrix associated with the inverse operation. Then E is invertible i.e.
Suppose that we’ve got two matrices of the same size A and B. If we can reach B by applying a finite number of row operations to A then we call the two matrices row equivalent.
Note that this will also mean that we can reach A from B by applying the inverse operations in the reverse order
If any diagonal element is zero then the matrix is singular
UpperTraingular Matrix=
LowerTraingular Matrix =
Matrix is orthogonal if
Matrix is orthogonal if
Matrices and are orthogonal if
Matrix is said to be similar to matrix if can be expressed as
where is some nonsingular matrix
Matrix is unitary if
is the complex conjugate of
A Hermitian matrix (also called selfadjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose  i.e. the element in the row and column is equal to the complex conjugate of the element in the row and column, for all and
·
· Example:
· Entries on main diagonal are always real
· A symmetric matrix with all real entries is Hermitian
A Hermitian matrix with complex entries which is equal to negative of its own conjugate transpose  i.e the element in the row and column is equal to the negative of complex conjugate of the element in the row and column, for all and
·
· E.g.:
· Entries on main diagonal are always purely imaginary
· If is skewHermitian then raised to odd power is skewHermitian
· If is skewHermitian then raised to even power is skewHermitian
Symmetric 


SkewSymmetric 


Hermitian 


SkewHermitian 


ConjugateSymmetric 


ConjugateAntiSymmetric 

Same as SkewHermitian 
A matrix is idempotent if
· An idempotent matrix is diagonalizable
· Eigen values are 0 or 1
· Rank of an idempotent matrix = sum of its diagonal elements
A matrix is Involutory matrix if
· If is involutory then
· A Matrix of the form is always involutory.
A matrix that has exactly one entry of 1 in each row and column and zero elsewhere is called a permutation matrix.
· It is a representation of permutation of numbers
· Example: is permutation matrix of the permutation (1,3,2,4)
Determinant functions: The determinant function is a function that will associate a real number with a square matrix.
Permutation of integers: An arrangement of integers without repetition and omission.
Theorem 1: If A is square matrix then the determinant function is denoted by ‘det’ and is defined to be the sum of all the signed elementary products of A.
1.
2. {it may be equal in some cases but not so in general}
3.
4.
5.
6. … if A is a triangular matrix
7. If is the matrix that results from multiplying a row or column of by a scalar c, then
If is the matrix that results from interchanging two rows or two columns of then
If is the matrix that results from adding a multiple of one row of onto another row of or adding a multiple of one column of A onto another column of then



2) Using Cofactor. 


Rank is the highest order of a nonzero minor of the matrix.
· Apply elementary row transformation.
· No of nonzero rows = rank of the matrix.
· Find all the minors of the matrix.
· Find the order of nonzero minors.
· The highest order is the rank.
· From the norm of the matrix
· Rank = r
·
·
·
·
·
[ is conjugate transpose of , conjugate transpose is the adjoint of any matrix]
Matrix is said to be diagonalizable if there exist a nonsingular matrix such that
Diagonalization of is possible if and only if the Eigen vectors of are linearly independent.
Some application of Diagonalization:
1. Evaluation of powers of :
2. Evaluation of function over : If then
3. Not all matrices are diagonalizable but real symmetric matrix (RSM) are always diagonalizable.
4. Eigen vectors of RSM are always distinct
A matrix is symmetric if . For example A=
Skew Symmetric: . For example B=
The diagonal elements of a skew symmetric is always zero
Eigenvalues of a real skew symmetric matrix is either zero or purely imaginary
The normal form of a matrix is where the rank of the matrix
Step 1 : 


Step 2 : 

Apply ERT on and till becomes Row Echelon Matrix. This operation converts into and into 
Step 3 

Apply ERT on and till becomes a matrix in normal form. This operation converts into and into 
Step 4 

This form is 
So given a matrix of rank two other non singular matrices and can be found such that
= square matrix (
=Eigen vector
= Eigen value (scalar)
· Eigenvalues and eigenvectors will always occur in pairs.
·
· The set of all solutions to is called the eigenspace of corresponding to λ.
· Suppose that λ is an eigenvalue of the matrix A with corresponding Eigenvector.
Then if k is a positive integer is an eigenvalue of the matrix with corresponding Eigenvector.
·
· =
· If are eigenvectors of corresponding to the k distinct eigenvaluesthen they form a linearly independent set of vectors.
Suppose that is a square matrix and if there exists an invertible
matrix (of the same size as ) such that is a diagonal matrix then we call
diagonalizable and that P diagonalizes A .(Columns of P are the eigenvectors of A).
Suppose that A is an n× n nonsingular matrix, then the following are equivalent.
(a) A is diagonalizable.
(b) A has n linearly independent Eigenvectors, which forms the column of the matrix that diagonalizes A.
Suppose that A is an matrix and that A has n distinct eigenvalues, then A is diagonalizable.
are the Eigen values of a square matrix


Same as that of but Eigen vectors are different. 












· Only Eigen vectors of distinct Eigen values are linearly independent.
· For Diagonalization of a matrix its Eigen vectors should be linearly independent.
Every square matrix satisfies its characteristic equation.
· It is used in calculating the power of matrices (in place of direct matrix multiplication)
The above system of equation is
If B=0 system is Homogenous.
If system in NonHomogenous.
1. Cramer's Method
2. Augmented Matrix Method
3. LUDecomposition Method
Limit 
A function is said to have a limit at a point if 1. along any path 

Continuity 
A function is said to be continuous at a point if 1. along any path 2. 

Differentiability 
A function is said to be differentiable at a point if
1. is continuous at 2. Limit exist


Rolle's Theorem 
If 1. is continuous in 2. differentiable in 3. Then there exist a in the interval such that 

Mean Value Theorem 
If 1. is continuous in . 2. differentiable in
Then there exist a in the interval such that


Trigonometric 


Exponential 


Logarithmic 


Differentiation 


Series 


LHospital's Rule


If is continuous and derivable in and then there exist at least one such that
If is a function such that is continuous in the interval and exist in the interval then Taylor's theorem says that there exist a number in the interval and a positive integer such that


Cauchy Series 


Lagranges Series 


Maclaurin's Series 
and 

Convenient form of Taylor's Theorem
Taylor Series expansion of about the point (or in powers of ()
Total Derivative 

is a function of independent variables 
Total Derivative 

is a function of and is independent variable 
First order Derivative of an implicit function



Second order partial derivative of an implicit function 


Homogenous Function 
If homogenous function of degree satisfies the equation


Euler's Theorem for Homogenous function of degree 


and are two functions of and . If there is another function such that then and are said to be functionally dependent.
Test for Functional Dependence
If z= is a function of and then
Using the definition of total derivative
Leibnitz's Rule : It is a rule that gives order derivative of product of two functions.
From Integral Calculus 

General Leibnitz Rule 

Leibnitz Rule 


While choosing give preference to functions on the right side of the ILATE table. While choosing give preference to functions on the left side of the ILATE table. 




















































Length of an arc/curve 


Surface Area of an arc/ curve (obtained by rotating it about x (or y) axis) 


Single Integral 

1. It gives the length of the arc from . 2. It also gives the area between the arc and the from 
Double Integral 
Double Integral (Polar Coordinates)

1. It gives the area of the region D.
2. It also gives the volume of the region D 
Triple Integral 

It gives the volume of the region enclosed by D 
ODE means that there is only one independent variable in the equation
ORDER: order of ODE is the order of the highest derivative appearing in it.
DEGREE: power of the highest order
Order = 3
Degree=6
A Differential Equation where , the independent variable and , the dependent variable is said to be LINEAR if
1. and all its derivative are of degree one.
2. No product term of and(or) any of its derivative forms are present.
3. No transcedental functions of and(or) any of its derivative are present.
Note: Linearity of a DE depends only on the way the depenedent variable appears in the equation and is independent of the way the independent variable appears in it.
Linear ODE 
NonLinear ODE 






Type 
Functional Form 
Examples 
Variable Separable Type I 

1. 2. 3. 4.

Variable Separable Type II (requires substitution ) 
1. Substitute 2. Solve the equations on the lines of Variable Separable Type I

1. 2. 3. 4. 
Variable Separable Type III (Homogenous ODE reducible to Variable Separable Type III) 
1. 2. Substitute 3. Solve the equations on the lines of Variable Separable Type III
Note: should be homogenous of same degree

1. 2. 3. 4.

Variable Separable Type IV (Nonhomogenous ODE reducible to homogenous type reducible to Variable Separable Type I/II) 
1. is nonhomogenous
2. ,substitute and else
3. Solve
4. is homogenous
5. Substitute
6. Solve the equations on the lines of Variable Separable Type III and replace p by and finally and
Note: should be homogenous of same degree 
1.
2.
3.
4.

Exact Differential 
If
is called an exact differential is called an exact differential equation 
Solution to Exact Differential Equation 
where or

Exact DE 
Solution 


Steps to Solve NonExact Differential Equation
where

Step 1: Reduce NonExact DE to Exact DE using Integrating Factor (I.F)
Step 2: Solve the Exact DE. Solution is of the form
or

Type 
Integrating Factor 







and are homogenous of same degree



and are homogenous of same degree and







where and is given by solving


Rearranging such that some special group of terms are formed


A linear first order first degree DE of the form
This DE is known as Leibnitz Linear Equation 
1. Here and
2. Also
3. So IF =
4. Solution is


A nonlinear first order first degree DE of the form
where
This DE is known as Bernoulli's Equation

1. Substitute
2. Equation reduces to Linear PDE
3. Solve using the method described for linear first order DE 

A nonlinear first order higher degree DE of the form



A nonlinear first order higher degree DE of the form
This DE is known as Clairaut's Equation 


A nonlinear first order higher degree DE of the form
This DE is known as Lagrange's Equation


General Linear Higher Order DE 

Solution of Homogenous LSODE (Constant Coefficients) of the form
or using the notation

1. Characteristic Equation is 2. If and are the roots of the equation

Solution of Homogenous LSODE (Variable Coefficients) of the form
provided the complimentary functions are available
This method is called the method of variation of parameters. 
If the complimentary solution is then
1.
If
2. The particular solution is then given as
3.

Note: This method is suitable only for second order DE with variable coefficients. There is no general method for solving higher order DE with variable coefficients.
Solution of NonHomogenous LSODE (CC) of the form
1. Characteristic Equation is 2. If and are the roots of the equation
3. Evaluation of Particular Integral (P.I)
4. Solution is 
Solution of Homogenous LHODE (VC) of the form
or

1. Substitute
2. Substitute
where
3. Now the equation is of Homogenous LSODE(CC) form with characteristic equation of the form
4. Solve the new Homogenous LHODE(CC) by finding C.F and PI and replace in the final solution 
Solution of NonHomogenous LHODE (VC) of the form
or
This equation is also known as CauchyEuler's Equation 
1. Substitute
2. Substitute
where
3. Now the equation is of NonHomogenous LSODE(CC) form with characteristic equation of the form
4. Solve the new Homogenous LHODE(CC) by finding C.F and PI and replace in the final solution 
Solution of NonHomogenous LHODE (VC) of the form
or
This equation is also known as Legendre's Equation 
1. Substitute
2. Substitute
where
3. Now the equation is of NonHomogenous LSODE(CC) form with characteristic equation of the form
4. Solve the new Homogenous LHODE(CC) by finding C.F and PI and replace in the final solution 
There is no general procedure for finding solutions to linear higher order DE with variable coefficients.
Some Special Forms 
How to approach for the solution 

Solve by repeated integration 

Multiply both sides by which makes the equation exact.
The solution is given by

Equations not explicitly containing x 

Equations not explicitly containing y 

Change of independent variable 

One Dimensional Heat Equation 
Boundary Condition 
Solution 

Homogenous Boundary Condtion



NonHomogenous Boundary Condtion



Homogenous with both ends insulated


A function is said to be analytic at if is differentiable at and in its neighbourhood.
is analytic is differentiable is continuous, but
the inverse is not always true.
Properties of analytic function 

If is analytic it satisfies CauchyRiemann Equation 

If is analytic then is also analytic 

If is analytic then and is harmonic 

Milne's Thompson Method :
If is analytic then can be evaluated by integrating after substituting and in i.e.


Cauchy Integral Theorem If is analytic in the simply connected Domain D and C is a closed curve in D then


Cauchy Integral Formula If is analytic in the simply connected Domain D and C is a closed curve in D enclosing point

Residue Theorem
If is analytic in the simply connected Domain D and C is a closed curve containing singularities then
Residue at

Taylor Series Theorem
If (z) is analytic in the region with center , then it can be uniquely represented by a convergent power series.

Laurent Series Theorem
If is analytic in the annulus region with center then it can be uniquely represented by a convergent power series known as Laurent Series

1. A sequence of complex numbers is an assignment of a positive integer to a complex number
2. A series of complete numbers is the sum of the terms in the sequence. For eg
3. Power Series: A power series in is defined as
4. Power series represent analytic functions. Conversely every analytic function can be represented as a power series known as the Taylor series. Moreover a function can be expanded about a singular point as a Laurent series containing both positive and negative integer powers of
If an event can be done in ways and another event can be done in ways then both the events can be done in ways provided they cannot be done simultaneously.
If an event can happen in ways and another event can happen in ways then in the same order they can happen in ways, provided they do not happen simultaneously.
· Permutation of a set of distinct objects is an ordered arrangement of these objects.
· Permutation of a set of distinct objects taken at a time is
· Permutation of r objects from a set of n objects with repetition is
· Combination of a set of distinct objects is an unordered arrangement of these objects.
· Combination of a set of distinct objects taken at a time is
· Combination of r objects from a set of n objects with repetition is
· If A and B are mutually exclusive then
· If A and B are mutually exclusive then
·
·
· if A and B are independent events.
Let the theorem of total probability gives
X is a random variable (in fact it is a single valued function) which maps s from the sample space S to a real number x.
· are different ways of writing the probability of
· f(x) is called the density function.
·
·
·
·
· F(x) is called the distribution function.
·
· Mean
· Variance
· Variance
·
· Probability that X will assume a value within k standard deviations of the mean is at least
·
·
· Binomial Distribution
· Mean = np=nq
· Variance = npq
· Hypergeometric Distribution
· Mean =
· Variance =
· Poisson's Distribution
·
· Mean =
· Variance =
· Normal Distribution
·
Bisection Method 

. . .

RegulaFalsi
(or Method of chords) 

When ├ f(a)>f(b)

NewtonRaphson Method
(or Method of tangents) 


A diagonal system is one in which in each equation the coefficient of a different unknown is greater in absolute value than the sum of the absolute values of coefficients of other unknown. For example the system of equations given below is diagonal.
GaussSeidel method converges quickly if the system is diagonal.
Steps in GaussSeidel method
1. Rewrite the equation as
2. Assume an initial solution (0,0,0)
3. Evaluate using Now use the new and to evaluate .
4. Now use and new to evaluate
5. Iterate these steps using new calculated values till desired approximation is reached.
6. If the system is not diagonal, GaussSeidel method may or may not converge.
Forward difference is defined as
·
·




























































Backward difference is defined as




























































x 
y 


















































































































































































Divided difference is defined as
For example
a.
b.
· Divided difference formula is used for interpolating data sets where the independent data points are unequally spaced i.e.
· But when the points are equally spaced i.e. it reduces to forward/backward difference formula
Examples:
1.
2.


















































































































h= is the uniform distance betwen the data points.
q= a variable introduced to simplify expressions.










































. . . 
. . . 
















h= is the uniform distance betwen the data points.
q= a variable introduced to simplify expressions.
Newton's divided difference interpolation formula






















































Gaussian Interpolation Formula
x 
y 




























































































































































































Gaussian Interpolation Formula I
Gaussian Interpolation Formula II
Gaussian Interpolation Formula III
Stirling Interpolation Formula (Arithmetic mean of and
Bessel's Interpolation Formula (Arithmetic mean of and






... 







... 

Lagranges interpolation formula is given by
For the given set of data points Lagranges polynomial can be written as








Lagranges inverse interpolation formula is
In spline interpolation a polynomial called spline S(x) is assigned to each sub interval and




... 




... 
Linear Spline 

Discontinuous first derivate at the inner knots. 
Quadratic Spline 

Continuous first derivate at the inner knots.
End points are connected with straight lines. 
Cubic Spline 

Continuous first and second derivate at the inner knots.
End points are connected with curves 
So a set of n+1 data points with n subintervals would have a different spline polynomial say for each subinterval.
Differentiating the polynomial formed using forward interpolation gives the differentiated polynomial
With backward interpolation
The error between interpolated values and actual values is large in case of differentiation than for a polynomial function.




... 





... 

Trapezoidal Rule (n=1) 

Simpsons Rule (n=2) 

Simpson's Rule (n=3) 

Weddle's Rule (n=6) 

Boole's Rule 

In all the above methods the general rule is to replace the tabulated points with a polynomial function and then carry out its integration. In other words the polynomial is an interpolation function evaluated using Newton's forward (or backward ) interpolation formula. Now if there are large data points then the polynomial becomes oscillatory.
Piecewise interpolation is about finding a separate polynomial for each subinterval rather than for the whole set of data points. These functions are called splines.
1. Taylor's power series method
2. Euler's method
3. Modified Euler's method
4. RungeKutta 4^{th} order method
1. Milne's predictor corrector method
2. Picard's successive approximation method
3. AdamBashforthMoulton method
Taylor series about a point is given by
Solve Given
Sol : From the given equation we have
Using Taylor's series about we have
Substituting we get the solution
· Euler's method does not give us analytical expression of y in terms of x, but it gives the value of y at any point x.
Solve . Given and use step size of and find
Sol:

x_{i} 



0 
0 
100 
1 
75 
1 
25 
75 
0.75 
56.25 
2 
50 
56.25 
0.5625 
42.1875 
3 
75 
42.18 
04218 
31.63 
4 
100 
31.63 


Predictor Formula :
Corrector Formula :
Solve . Given , use step size of Find
Sol:

x_{i} 





0 
0 
100 
1 
75 
0.75 
78.125 
1 
25 
78.125 
0.5859 
58.59 
0.5859 
61.034 
2 
50 
61.034 
0.6103 
45.77 
0.4577 
47.68 
3 
75 
47.68 
0.4768 
35.76 
0.3576 
37.25 
4 
100 
37.25 




Predictor Formula
Corrector Formula
.
.
.
and so on till desired approximation is achieved
Adam Bashforth's Predictor formula
Adams Moulton's Corrector formula
It is a nonselfstarting four step method which uses 4 initial points to calculate for . It requires only two function evaluation of per step.
Circle

Ellipse

Segment of a line from point to


· Line integral converted to double integral
· Line integral converted to surface integral and viceversa
A vector field is called conservative if a scalar function can be found such that . The scalar function is called potential function of .
· Surface integral and Volume integral
Line integral w.r.t to arc length does not change sign if curve is traversed in the opposite direction 

Line integrals w.r.t variables or change sign if curve is traversed in the opposite direction 

Given and
The line integral of vector field is defined as
A vector normal to the curve is given by
· if where is a scalar function
where the surface S is and D is the region for double integration in the plane.
1. Evaluate a unit normal vector to the elemental surface . This converts the integral
2. Projection* of onto the plane
3. Substitution for
4. After all the above substitution we have
5. Evaluate the multiple integral

Evaluate where and is plane located in the first quadrant.
1. A vector normal to is
2. Projection of on axis
3. After substitution
4. Substitute
5. Evaluate the double integral

* The concept of projection of one vector over another is used here.
1. 
Abort execution of a command 
Alt+. 
2. 
Abort execution of a command 
Evaluation > Quit Kernel > Local 
3. 
Include a comment 
* Insert comments between asterisk * 
4. 
TraditionalForm[] 

5. 
Symbol=. 
Removes the value of the symbol 
6. 
Expression//Command 
It is the same as Command[Expression] 
7. 
N[Expression,n] 
attempts to give answer in n decimal digits 
8. 
?Command 
Gives help about the command 
9. 
??Command 
Gives help on the attributes and options of the command 
10. 
Ctrl+K 
Shows all command starting with say ‘Arc’ 
11. 
?Command* 
Shows all command beginning with say ‘Arc’ 
12. 
?`* 
Lists all global variables 
13. 
/. 
Replacement or Substitution 
14. 
/; 
Conditional 
15. 
Together[] 
Combines the difference or sum of fractions 
16. 

Function argument on the left hand side is always suffixed with an underscore in user defined functions 
17. 
:> or :> 
Rule Delayed 
18 
# & 
Pure function 
19. 
/@ 
Used to map a function to a list Map[f,list] f/@list 
Using: = is a must. The argument of the function is suffixed with an underscore.
It is better to use Piecewise [ ] function to create such functions because limits and continuity can be then be checked.
Expression//MatrixForm
Expression//TableForm
Expression//TraditionalForm
NestList[] 

Range[] 

Array[] 

Table[] 

1. 
Solve [equation, variables] 

2. 
Reduce[equation, variables] 

3. 
Solve[equation, variables]//ComplexExpand 
Represents complex number in traditional form rather than as rational power 
4. 
Eliminate [equation, variables] 
Removes variable from a set of simultaneous equation 
5. 
NSolve [ equation, variables] 
Gives numerical solution 
6. 
FindRoot [equation, startingvalue] 
Gives solution to ‘transcendental equation’ 
7. 
LinearSolve [a,b] 
Produces vector such that 