Double integral

Introduction to double integral:Let us consider a single valued function f(x, y) of two independent variables x and y defined in a closed region R of the x, y plane. Divide the region R into n sub regions ΔR₁, ΔR_{2, ....} ΔR_n by drawing lines parallel to the axes of coordinates. 

For solving problems on double integral,you will need to know the integrals formula.


Double integral polar co-ordinates: In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.

Analytical Geometry

 Analytic geometry is the study of geometry using a coordinate system and the principles of algebra and analysis.It is also known as co-ordinate geometry or cartesian geometry.

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and square often in two and sometimes in three dimensions of measurement .

Analytical geometry basics
Points, Straight lines, Planes, Second-order curves and Surfaces are the basics of analytical geometry.

Straight lines are  formed when minimum of two points is joined with a constant direction.Planes are formed by two dimensional surface.The dimensional surface is formed by infinite number of points. Analytical geometry is concerned with all kinds of surfaces and curves like conic section.

Pre algebra

Pre-algebra is a course in Math .It is generally taught to students in the 7th and 8th standard.Pre-algebra is taught to students as a preparation for learning algebra.

Pre-algebra includes learning about natural numbers, integers, fractions, decimals, factorization of natural numbers, etc.

You can get pre-algebra help online.You can avail online tutors at home by sitting in front of PC.

Pre- Algebra requires a clear understanding of concepts and  high school Algebra is based on it.

Graph Theory

In Graph Theory, a graph is an accumulation of dots that may or may not be joined to  each  other by lines. It doesn't amount how big the dots are, how long the curves are, or whether the curve are straight, curved, or squiggly. The "dots" needs not to be round! 

All that matters is which dots are joined by which lines.

Two dots can alone be affiliated by one line. If two dots are joined by a line, it's not "legal" to draw another line joining them, even if that line stretches far  from the first one
 Graph theory applications: Graph theory  plays an important role in the design, analysis, and testing of computer programs. It is important for the fact that flow of control and flow of data for any program can be expressed in terms of directed graphs.

Geometry constructions

Construction of geometry involves drawing of shapes.In construction, we use compass and scale. 

Geometry construction rotation: Circling is the movement (turning) of an object about a point through a given  number of degrees in a clockwise or an anticlockwise. A two-dimensional object rotates in the adjustment of a center most or position of rotation.

See the link for Geometry Construction Answers.

Example of construction of geometry:

Construction of parallelogram type of quadrilateral is AB = 11 cm, BC = 10 cm and ÐA = 60o

Solution:

First we can draw AB - 11 cm and make  ÐA = 60o

Then cut the line AD - 10 cm ( accordingly AD = BC)With B as center and a radius 10 cm, draw one arc and with D as center and radius11 cm cut that arc at C.

Join the line BC and DC.Then ABCD is the construction of appropriate parallelogram.

Matrices

What are matrices?
    A rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular arrangement of entries are enclosed by bracket or by square bracket.

You can also check the link on matrices determinants and determinants of matrices.

There are different types of matrices -  Row Matrix, Column Matrix, Square Matrix, Diagonal Matrix, Scalar Matrix, Identity or Unit Matrix, Null Matrix or Zero Matrix

Operation on matrices:
Equality of Matrices, Addition of Matrices, Matrix Addition is commutative, Matrix addition is associative, Subtraction of Matrices, Multiplication of a matrix by a scalar, Multiplication of Matrices, Properties of Matrix Multiplication, Transpose of a Matrix, Properties of Transpose, Symmetric Matrix, Skew-Symmetric Matrix, Properties of Symmetric and Skew Symmetric Matrices,

Application of matrices:Matrices are applied on Homogeneous Equations (Constant = 0), Non Homogenous Equations.

Mathematical Induction

Mathematical Induction:It is a technique for proving a statement, a theorem, or a formula that is asserted about every natural number.

Before we take an example on mathematical induction I would also like to provide you a link on Mathematical reasoning. 

Example of mathematical induction:

Mathematical induction is used to prove that the following statement holds for all natural numbers n.

Let us call the statement as P(n)

Show that the statement holds for n = 0.
P(0) amounts to the statement:

On the left-hand side of the equation, 0 is the only term, and so the left-hand side is equal to 0.
On the right-hand side of the equation, 0·(0 + 1)/2 = 0.
The two sides are equal, so the statement is true for n = 0. So, it has been shown that P(0) holds.

Inductive step: Prove that  if P(n) holds, then  P(n + 1) also holds.

Imagine P(n) holds (for some unspecific value of n). It must  be proved that P(n + 1) holds, i.e,

According to the induction hypothesis that P(n) holds, we can rewrite  the left-hand side as follows:

Algebraically:

Hence P(n + 1) holds.

We have proved both the  basis and the inductive step, therefore,it has been proved by mathematical induction that P(n) holds for all natural n