Divergence theorem

Divergence theorem commonly known as divergence theorem of gauss. This theorem also help to integrate the functions, divergence theorem is a combination of vector and integral. To understand the theorem we have to know about divergence of vector field.
Vector calculus is very important in engineering and physics to the gradient, divergence and curl. Lets take vector V(x,y,z), be a differential vector function where x y z are Cartesian coordinate and if v1,v2,v3 are component of vector V then the function div.V is called divergence of vector field. The physical meaning of divergence is value of a function that characterize a physical or geometrical property must be independent of the particular choice of coordinate that means those values must be invariant with respect to coordinate transformation.
Now divergence-theorem in integrals, this theorem is used for triple integration. Divergence-theorem, which transforms surface integral to triple integral, it also involves the divergence of vector function. The triple integration is a generalization the double integration. triple integral can be transform in to surface integral, over the boundary surface of a region in space and conversely. This is a practical interest because one of the two kinds of integral is often simpler than other. It also helps in establishing fundamental equation in fluid flow, heat conduction etc. the transformation is done by gauss divergence-theorem which include the divergence of a vector field function.
Divergence-theorem of gauss means also transformation between volume integral and surface integral. Let constant T be a closed bounded region in space whose boundary is a piece wise smooth Orientale surface and let F(x,y,z) be a vector function that is continuous and continuous first partial derivatives in some domain containing T. then we proof the divergence-theorem of gauss. In this theorem a unit normal vector of surface which is pointing to the out side of surface can also used.
Let’s clear this theorem with some Divergence Theorem Examples. Suppose we have to evaluate the surface integral by this divergence-theorem. In our problem surface S is a closed surface consisting cylindrical coordinates and also coordinates of circular disc are given.  To solve this problem we need to know triple integration. Then we can solve this type of problem, when we solve this types of problem we mostly use polar coordinates, which is defined by X=rcosθ and y = rsin θ
Let’s take one more Divergence Theorem Example that is transformation of volume integral in to surface integral. Suppose function which is in vector form is given and function also involve unit normal vector. Here in place of surface volume of sphere is given, now by using gauss divergence-theorem we integrate the function from volume integral to surface integral.

Introduction to algebra test questions

Algebra is one of a part of arithmetic that uses letters in place of some unknown numbers. It is concerning about learn the rules of operations and dealings, and the constructions. It also was including the conditions, polynomial expressions, equations and algebraic arrangement. In other words, we use numerals like 1, 2, 3, etc to represent numbers. In algebra we use numbers as well as letters of the alphabet such as a, b, c, etc for any numerical values we choose. In this article we shall discuss about algebra test questions.First we will solve some sample questions for algebra test and then do some practice questions.

Sample Questions for Algebra Test :

Ex 1: Solve the linear questions

19x - 7 = 21x – 9

Solution: Subtract 21 x from both sides of the above equation, we get the below term
            -2x – 7 = -9
Add 7 to both sides of the above equation, we get the below term

-2x – 7 + 7 = - 9 + 7

-2x = - 2

Divide both sides by -2 of the equation, we get  

        X = 1

The answer is x = 1

Test the answer for the given equation. Replace with in the above given equation for x. If the left side of the equation sum value is equal to the right side of the equation sum value after the replacement of x value, you have got the exact answer.

Left side of the equation:

 19(1) - 7 = 12

Right side of the equation:

 21(1) - 9 = 12

Ex 2: Solve the given Questions

12− (3x + 7) = 2x

Solution: In the given equation is expanding and the terms are out the bracket.

12 − (3x + 7) = 2x

12 −3 x − 7 = 2x

(12 – 7) − 3x = 2x

5 – 3x = 2x

Now we are identify that it is easier to get all the x value on the right side of the given equation, by adding x to both sides of the above equation.

5 = 5x

Now we divide both sides by 5 and swap the sides:

We get the x value is 1

Test the answer for the given equation. Replace with in the above given equation for x.

We get

12 – 3(1) + 7 = 2(1)

12 – 3 - 7 = 2

Practice Questions for Algebra Test :

Solve 17 x - 6 = 23 x – 18
                    Ans: x = 2

Solve 13− (5x + 9) = -3x
                    Ans: x = 2

Intermediate algebra test

Intermediate algebra is mainly used for solving the equations. Intermediate algebra mainly uses the simplification technique for solving the equations. Intermediate algebra uses the letters and variables. Intermediate algebra also deals with the expressions and exponents of polynomials. Algebra test problems are very simple and easy. Intermediate algebra test covers the topics,

• linear Equations
• Equation of a line
• Quadratic Equations
• Polynomial Functions
• Slope formula

Intermediate Algebra Test Problems

Algebra test Problem: 1

Solve the equation x + y = 6, x - 3y = 2.

A) x = 2, y = 4 B) x = 5, y = 2 C) x = 4, y = 2 D) x = 5, y = 1.

Answer: D

Algebra test Problem: 2

Find the slope of the equation 4x + 3y = 21

A) M = - 4 / 5 B) M = - 4 / 3 C) M = - 3 / 4 D) M = - 3 / 5.

Answer: B

Algebra test Problem: 3

Factorize the equation 2x2 - 5x + 3.

A) X = 2, X = 4 B) X = - 2, X = - 3 C) X = 1, X = 3 / 2 D) X = 1, X = 2 / 3.

Answer: C

Algebra test Problem: 4

Find the vertex of the equation Y = x2 + 4x + 7

A) (2, 4) B) ( -2, 3) C) (-3, 2) D) (1,-1).

Answer: B

Algebra test Problem: 5

Find the distance between the two points (1, 5) and (3, 9).

A) d = 6 B) d = 3 C) d = 7 D) d = 5.

Answer: D

Problems for Intermediate Algebra Test

Algebra test Problem: 6

 Find the value of x for the expression: 4x + (4 - 7x) = (2x + 3) - 4.

A) 3 B) 2 C) 8 D) 1

Answer: D

Algebra test Problem: 7

Add the two expressions (3x + 2y - 8) and (5x - 3y -4)

A) 5x + 4y - 12 B) 8x + y + 12 C) 8x - y - 12 D) 2x - y - 4.

Answer: C

Algebra test Problem: 8

The sum of the number and 32 is 67. Find the unknown number.

A) 32 B) 33 C) 31 D) 35

Answer: D

Algebra test Problem: 9

Multiply (3a2 + 4b) (2a + 6b)

A) 6a3 + 8ab + 18a2b + 24b2

B) 3a3 + 8ab + 18a2b + 24b2

C) 4a3 + 6ab + 16a2b + 24b2

D) 12a3 + 9ab +18a2b + 24b2

Answer: A

Solving Trigonometric Equations made easy

The six trigonometric functions are the sine, cos, tan, csc, sec and cot. An equation consisting of one or more trigonometric functions is called a trigonometric equation, for instance cot^2(x) + 1 = csc^2(x) is a trigonometric equation.  Some of Trigonometric Equations Examples are 2 cos^2(x) –sin(x) – 1 =0, 2sin(theta) – 1 = 0, cos(theta/2) = 1 + cos(theta), 6 sin^2(theta) – sin(theta) = 1, cot(2 theta) – tan(2 theta) = 0

Quadratic Trigonometric Equations
Trigonometric equations that involve trigonometric functions as products are called the quadratic trigonometric functions. In a quadratic equation with degree 2 written in the form of  ax^2 + bx + c. So, a quadratic trigonometric equation will consist of trigonometric functions with degree 2. For instance, 2 sin^2(x) – sin(x) +1 = 0 is an example for a quadratic trigonometric equation.

Let us now learn the steps involved in simplifying trigonometric equations. We solve trigonometric equations as we solve any other algebraic equation except that we use the trigonometric identities in solving. They are also solved by the sign of the trigonometric value by determining the quadrant(s) for that particular trigonometric value. Let us simplify a trigonometric equation, 2 sin(pi/2 -alpha) =1. We shall use the identity sin (pi/2 – alpha) = cos(alpha) in solving the given equation. So, the trigonometric equation reduces to, cos(alpha)=1/2 . Now we need to check for what angle of cosine alpha gives the value ½. We know that cos(pi/3) equals ½. So, we write it as cos(+/-) pi/3 =1/2. The positive and negative sign depends  on which quadrant the angle alpha lies and hence we can generalize the solution as alpha = (+/-)pi/3 + 2k pi where k is any integer. There is a significance for adding 2k pi to the solution, as per the periodic behavior of the function cosine. So, the solution set is written as {alpha = (+/-)pi/3 + 2k pi: k belongs to set of integers}

Sin(x) + cos(x) = 0
Simplifying the above equation we get, sin(x) = -cos(x). Here, the value of sine and cosine have to be equal with opposite sign. Let us write sine in terms of cosine using trigonometric identity, which gives us
  cos(pi/2 – x+2 k pi) = cos(pi + x)
      pi/2 – x + 2 k pi = pi + x

The general solution would be, x = -pi/4 + k pi where k belongs to set of integers
Now, in the interval [0, 2pi) let us consider k = 1 and k = 2, we get, x = (3/4) pi and x = (7/4) pi as the solution that is at 135 degrees and 315 degrees the value of sine and cosine are equal and opposite.

Steps for solving standard deviation problems

A standard deviation example would typically have a set of observations x_1, x_2, x_3,  …… x_n. The number of observations being n. In general it can be written like this: x_i, where i = 1, 2, 3, … n. To calculate the standard deviation of such a problem the following steps are followed:

Step 1: List all the observations in the first column of a table. The title of column would be x_i and the observations would be x_1, x_2, x_3,  …… x_n.

Step 2: Find the mean of the observations. If we denote the mean by x ¯, then the formula for calculating the sample mean would be:  x ¯ = (x_1+ x_2 + x_3+  ……+ x_n)/n = [∑x_i]/n

Step 3: We now come to the second column. In this column we find the deviation of each of the observation from the mean. For that we calculate the value of x_i - x ¯ for each of the observations.
Note that in this step it is possible that we get negative values. We shall take care of the negatives in the next step.

Step 4: Now we construct the third column. In this column we square the results of column (3) for each of the observations. That means we find the value of (x_i - x ¯)^2 for each of the n observations. Note here that since we squared the difference, the negatives are got rid of so we need not worry about the negative numbers canceling out the positive numbers and affecting our end result.

Step 5: Now we find the standard deviation for each of the observation. For that we compute the following value: (x_i - x ¯)^2 / n. That would be placed in our fourth column.

Step 6: Adding standard deviations for each of the observations we get the value of the expression : ∑((x_i - x ¯)^2 / n). That will give us our sample standard deviation formula. The symbol for standard deviation is the Greek small alphabet ‘s’ pronounced as ‘sigma’. Therefore we say that:
s = sqrt[∑((x_i - x ¯)^2 / n)] or s = sqrt [∑(x_i - x ¯)^2/n]

Inverse Functions and their derivatives in a nutshell

In Math, Inverse Function is a function whose relation to a given function is such that their composite is the identity function. Inverse Function can also be defined as the function obtained by expressing the dependent variable of one function as the independent variable of another; for instance f and g are inverse-functions if f(x)=y and f(y)=x. It is denoted by (-1) in the power, inverse of f(x) is f-1(x).

One important point we need to remember is that the inverse of a function may not always be a function. While finding Inverse of Function of a function y in terms of x, we just switch the x and y and then solve for y. The new function [y-1] we get is the inverse-function of the given original function. For instance, finding Inverse of a Function f(x)= (x/3) -1 involves a list of steps. First let us take f(x)=y=(x/3) -1. In the next step we switch the x and y, that gives us, x = (y/3)-1. The last step would be to solve for y, finally we get y= y-1= 3x+3 which is the inverse-function of f(x) = (x/3) -1

There are four main steps involved in Solving Inverse Functions. Given a function f(x) in terms of x,
1. First step, we write the given function f(x) equal to y
2. Second step involves interchanging the x and y
3. In the third step, we solve for y
4. Fourth step, we write y as f-1(x) which is the inverse-function of f(x)

Considering an example, let us now learn how to solve Inverse functions using the above steps. Given the function f(x) = 2x-3, let us solve inverse-function [f-1(x)]. The first step would be to write y = 2x-3. Next we interchange x and y which gives, x = 2y-3. Now, we solve for y; y = (x+3)/2. The last step is to put f-1(x) in place of y which gives us f-1(x)= (x+3)/2  which is the inverse-function of the given function.

Derivative of an Inverse Function, if for a function f an inverse-function f-1 exists and if f is differentiable at f-1(x) and f’[f-1(x)] is not equal to zero, then f-1 is differentiable at x and the formula is given as:

Considering an example, let us now learn how to solve Inverse functions using the above steps. Given the function f(x) = 2x-3, let us solve inverse-function [f-1(x)]. The first step would be to write y = 2x-3. Next we interchange x and y which gives, x = 2y-3. Now, we solve for y; y = (x+3)/2. The last step is to put f-1(x) in place of y which gives us f-1(x)= (x+3)/2  which is the inverse-function of the given function.

Derivative of an Inverse Function, if for a function f an inverse-function f-1 exists and if f is differentiable at f-1(x) and f’[f-1(x)] is not equal to zero, then f-1 is differentiable at x and the formula is given as:

A square root function is the inverse-function of a square function f(x) = x2. So, the derivative of an Inverse Function of square function f(x) = x2 can be solved using the power rule

An Introduction to Quartiles in Statistics

Quartile is a term derived from the word ‘quarter’ which is one fourth of something. So, we can say  Quartiles Statistics is a certain fourth of a data set. When a given data is arranged in the ascending order that is from the lowest value to the highest value and this data is divided into groups of four, we get what we call Statistics Quartiles.

In Quartiles Statistics there are three quartiles which are, the first quartile also called as lower quartile is denoted as Q1 which lies in the twenty five percent of the bottom data. The second quartile is the median of the data that divides the data in the middle and has fifty percent of the data below it and the other fifty percent of the data above it. It is denoted as Q2.The third quartile also called the upper quartile denoted as Q3 has seventy five percent of the data below it and twenty five percent of the data above it.

We can state the Quartiles Definition as, Quartiles are the three values that divide an ordered data set into four approximately equal parts. Definition of Quartiles can also be given as, A statistical term describing a division of a data set into four defined intervals based upon the values of the data and how they compare the entire data set.

Now that we are a bit familiar with Quartiles Statistics, let us learn how to calculate Quartiles of a given data set. For a better understanding of the steps involved let us consider a data set. Calculate the three
 quartiles of the data set of scores,  27 18 20 20 23 29 24.

The steps involved are:
• Count the total number of scores in the given data; n= 7
• Arrange the data set of scores in the ascending order, that is, from the lowest score to the highest score, we get, 18 20, 20, 23, 24, 27, 29
• Find the median (Q2) which is the middle value of the data set
Median = ½(n+1) = ½(7+1)= 4
So, the median (Q2)=23 [the fourth term of the ordered data set]
• Next we shall calculate the lower quartile (Q1) which is the median of the lower half of the data set.    Q1 = ¼(n+1) = ¼(7+1) = 2
So, the lower quartile (Q1) = 20 [the second term of the ordered data set]
• Finally we shall calculate the upper quartile (Q3) which is the median of the upper half of the data set. Q3=3/4(n+1) = ¾(7+1) = 6
So, the upper quartile (Q3)= 27 [the sixth term of the ordered data set]

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