1) If A is a matrix then the reduced row-echelon 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 may-not 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.

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

LU DECOMPOSITION

SPECIAL MATRICES

Diagonal Matrix

If any diagonal element is zero then the matrix is singular

Upper-Traingular Matrix=

Lower-Traingular Matrix =

ORTHOGONAL MATRIX

Matrix is orthogonal if

Matrices and are orthogonal if

SIMILAR MATRIX

Matrix is said to be similar to matrix if can be expressed as

where is some non-singular matrix

UNITARY MATRIX

Matrix is unitary if

is the complex conjugate of

HERMITIAN MATRIX

A Hermitian matrix (also called self-adjoint 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

Skew-Hermitian Matrix

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 skew-Hermitian then raised to odd power is skew-Hermitian

· If is skew-Hermitian then raised to even power is skew-Hermitian

Symmetric
Skew-Symmetric
Hermitian
Skew-Hermitian
Conjugate-Symmetric
Conjugate-Anti-Symmetric		Same as Skew-Hermitian

IDEMPOTENT MATRIX

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

INVOLUTORY MATRIX

A matrix is Involutory matrix if

· If is involutory then

· A Matrix of the form is always involutory.

PERMUTATION MATRIX

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)

DETERMINANTS

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.

Properties of Determinants:

2. {it may be equal in some cases but not so in general}

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

Signed Product

INVERSE

Methods of finding Inverse of a matrix:

1) Using Elementary Row-Transformation
2) Using Cofactor.

RANK

Rank is the highest order of a non-zero minor of the matrix.

Methods to find the rank of a matrix

Method-1

· Apply elementary row transformation.

· No of non-zero rows = rank of the matrix.

Method-2

· Find all the minors of the matrix.

· Find the order of non-zero minors.

· The highest order is the rank.

Method-3

· From the norm of the matrix

· Rank = r

Properties of Rank

[ is conjugate transpose of , conjugate transpose is the adjoint of any matrix]

MINOR & COFACTOR

DIAGONALISATION

Matrix is said to be diagonalizable if there exist a non-singular 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

SYMMETRIC MATRIX

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

NORMAL FORM

The normal form of a matrix is where the rank of the matrix

Procedure to find the Normal form of a 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

EIGEN VALUE & VECTOR

= 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.

DIAGONALIZATION:

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 eigen-vectors of A).

THEOREM 1:

Suppose that A is an n× n non-singular matrix, then the following are equivalent.

(a) A is diagonalizable.

(b) A has n linearly independent Eigen-vectors, which forms the column of the matrix that diagonalizes A.

THEOREM 2:

Suppose that A is an matrix and that A has n distinct eigenvalues, then A is diagonalizable.

Properties of Eigen Values

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.

CAYLEY-HAMILTON

Cayley-Hamilton Theorem:

Every square matrix satisfies its characteristic equation.

· It is used in calculating the power of matrices (in place of direct matrix multiplication)

SYSTEM OF EQUATIONS

The above system of equation is

If B=0 system is Homogenous.

If system in Non-Homogenous.

Methods of solving

1. Cramer's Method

2. Augmented Matrix Method

3. LU-Decomposition Method

CALCULUS

Continuity

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

Common Limits

Trigonometric

Exponential

Logarithmic

Differentiation

Series

LHospital's Rule

Form	Convert To Form
	not required

Rolle's Theorem

If is continuous and derivable in and then there exist at least one such that

Machine generated alternative text:
R7)
I
. - -- () o
Ç(4) J(b) t
I I
I I
I I
a !
b

Differentiation

Taylor's Theorem

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

Maclaurin'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

Derivative of Implicit Function

First order Derivative of an implicit function

Second order partial derivative of an implicit function

Euler's Theorem (Homogenous functions)

Homogenous Function

If homogenous function of degree satisfies the equation

Euler's Theorem for Homogenous function of degree

Functional Dependence

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

Approximation

If z= is a function of and then

Using the definition of total derivative

Differentiate using Leibnitz's Rule

Leibnitz's Rule : It is a rule that gives order derivative of product of two functions.

From Integral Calculus
General Leibnitz Rule
Leibnitz Rule

Integration

	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.

Trigonometric Functions

Common Trigonometric Functions

Common Transformations

Application of Integration

Length of an arc/curve

Machine generated alternative text:
y = f(x)
b

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

Machine generated alternative text:
y = f(x)
b

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

ORDINARY DIFFERENTIAL EQUATIONS

ODE means that there is only one independent variable in the equation

ORDER & DEGREE

ORDER: order of ODE is the order of the highest derivative appearing in it.

DEGREE: power of the highest order

Order = 3

Degree=6

Understanding Linearity of ODE

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	Non-Linear ODE

Variable Separable Types

Type

Functional Form

Examples

Variable Separable Type I

Variable Separable Type II

(requires substitution )

1. Substitute

2. Solve the equations on the lines of Variable Separable Type I

Variable Separable Type III

(Homogenous ODE reducible to Variable Separable Type III)

2. Substitute

3. Solve the equations on the lines of Variable Separable Type III

Note: should be homogenous of same degree

Variable Separable Type IV (Non-homogenous ODE reducible to homogenous type reducible to Variable Separable Type I/II)

1. is non-homogenous

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

Exact Differential Equation

Exact Differential

is called an exact differential

is called an exact differential equation

Solution to Exact Differential Equation

where

Exact DE	Solution

Non Exact Differential Equation

Steps to Solve Non-Exact Differential Equation

where

Step 1: Reduce Non-Exact DE to Exact DE using Integrating Factor (I.F)

Step 2: Solve the Exact DE. Solution is of the form

Type

Integrating Factor

and are homogenous of same degree

and

where and is given by solving

Rearranging such that some special group of terms are formed

Group of Terms	IF	Exact Differential
	1





Group of Terms	IF	Exact Differential

Linear / Non-Linear First Order DE

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 non-linear 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 non-linear first order higher degree DE of the form

Solvable for	Solvable for	Solvable for

A non-linear first order higher degree DE of the form

This DE is known as Clairaut's Equation

A non-linear first order higher degree DE of the form

This DE is known as Lagrange's Equation

Linear Second/Higher Order DE (LODE)

General Linear Higher Order DE

Homogenous LODE

Non-Homogenous ODE

Constant coefficients LODE

or (LODE (CC))

Homogenous LSODE (CC)

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

Types of the root	Form of the Complimentary function
Real and distinct ( an
Real and same ( an
Complex conjugate and

Homogenous LSODE (VC)

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

The particular solution is then given as

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.

Non Homogenous LSODE (CC)

Solution of Non-Homogenous LSODE (CC) of the form

1. Characteristic Equation is

2. If and are the roots of the equation

Types of the root	Form of the Complimentary function
Real and distinct ( and
Real and same ( and
Complex conjugate and
Two Equal Complex conjugate roots

3. Evaluation of Particular Integral (P.I)

Type of	P.I
	when
	If the denominator is odd power of , apply conjugation to convert it to even powers of



is a general function

4. Solution is

Non Homogenous LHODE (VC) reducible to LHODE (CC)

Solution of Homogenous LHODE (VC) of the form

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 Non-Homogenous LHODE (VC) of the form

This equation is also known as Cauchy-Euler's Equation

1. Substitute

2. Substitute

where

3. Now the equation is of Non-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 Non-Homogenous LHODE (VC) of the form

This equation is also known as Legendre's Equation

1. Substitute

2. Substitute

where

3. Now the equation is of Non-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

General LHODE (VC)

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

Machine generated alternative text:
xL
f(L,t)=O f(L,c)=O
k
k”
f(Ot)O f(O,oo)=O
x=O
Initiol Fino!
Condition Condition

Homogenous Boundary

Condtion

Machine generated alternative text:
f(L,t)B0 F1
I’.I
loi
II
f(OM=Ao U ffQ,ao )=A1
. . x=O
Initio! Final
Conthtion Condition

Non-Homogenous

Boundary Condtion

Machine generated alternative text:
x’L
f’ (L,t)O f’ (L,»» )=O
k
o
.-
f’(Û,t)=O f(O,oe)=O
x=O
Initiol Final
Condition Condition

Homogenous with both ends insulated

COMPLEX FUNCTION THEORY

Analytic Function

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 Cauchy-Riemann 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.

Complex Integration Theorems

Cauchy Integral Theorem

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

Machine generated alternative text:
fiz) is analytic
inside C

Machine generated alternative text:
fiz) is analytic everywhere inside C
(z) = is analytic everywhere inside C except at z0
z — z0

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

Power 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

PROBABILITY

Probability Theorem

Principle of sum

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.

Principle of product

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

· 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

· 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

Probability

· If A and B are mutually exclusive then

Conditional Probability

· if A and B are independent events.

Theorem of Total Probability (Rule of Elimination)

Let the theorem of total probability gives

Bayes' Theorem

RANDOM VARIABLE

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.

Properties of Random variable

· Mean

· Variance

Chebyshev's Theorem

· Probability that X will assume a value within k standard deviations of the mean is at least

Distribution

· Binomial Distribution

· Mean = np=nq

· Variance = npq

· Hyper-geometric Distribution

· Mean =

· Variance =

· Poisson's Distribution

· Mean =

· Variance =

· Normal Distribution

NUMERICAL METHODS

Roots of Non-linear (Transcendental) Equations

Methods of solving non-linear equations / Finding roots of non-linear equations

Bisection Method

Regula-Falsi

(or Method of chords)

When ├ f(a)>f(b)

Newton-Raphson Method

(or Method of tangents)

Gauss-Seidel Method

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.

Gauss-Seidel method converges quickly if the system is diagonal.

Steps in Gauss-Seidel 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, Gauss-Seidel method may or may not converge.

Finite Differences

Forward Difference

Forward difference is defined as

Backward Difference

Backward difference is defined as

Central Difference Table

x	y

Divided Difference

Divided difference is defined as

For example

· 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:

Interpolation

Newton's Interpolation Formula

Newtons Forward Interpolation Formula

h= is the uniform distance betwen the data points.

q= a variable introduced to simplify expressions.

Newtons Backward Interpolation Formula








. . .	. . .

h= is the uniform distance betwen the data points.

q= a variable introduced to simplify expressions.

Newton's Divided Difference Formula

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

						...
						...

Lagranges interpolation formula is given by

For the given set of data points Lagranges polynomial can be written as

Lagranges inverse interpolation formula is

Spline Interpolation

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 sub-intervals would have a different spline polynomial say for each sub-interval.

Numerical Differentiation

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.

Numerical Integration

				...
				...

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.

Numerical Solutions of First Order ODE

Single-step methods

1. Taylor's power series method

2. Euler's method

3. Modified Euler's method

4. Runge-Kutta 4^th order method

Multi-step methods

1. Milne's predictor corrector method

2. Picard's successive approximation method

3. Adam-Bashforth-Moulton method

Taylor's power series 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

· 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

Modified Euler's Method

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

Runge-Kutta 4^th order method

Milne's Predictor Corrector method

Predictor Formula

Corrector Formula

Piccard's Successive Approximation

and so on till desired approximation is achieved

Adam-Bashforth-Moulton's Method (ABM)

Adam Bashforth's Predictor formula

Adams Moulton's Corrector formula

It is a non-self-starting four step method which uses 4 initial points to calculate for . It requires only two function evaluation of per step.

Numerical Solutions of PDE

VECTOR CALCULUS

Line & Surface Integrals

Machine generated alternative text:
Fdš
Divergence Theorem
Ë).d

Some common curves in parametric form

Circle

Ellipse

Segment of a line from point to

2. Projection* of onto the plane

Machine generated alternative text:
Proiechon ol ds on xv plane

3. Substitution for

Machine generated alternative text:
?

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

Machine generated alternative text:
3x.2y — lO
10— Jx

* The concept of projection of one vector over another is used here.

MATHEMATICA

Basics

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

Conditional Statements

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.

Matrix form

Expression//MatrixForm

Expression//TableForm

Expression//TraditionalForm

List generating Functions

NestList[]
Range[]
Array[]
Table[]

Algebraic Equation

Solving Algebraic Equation

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