How to find Directional Derivatives?

In calculus, the directional derivative is a differentiable function which multivariate along a vector V with a point P. It represents nothing but the rate of change of a function which moves in the direction of V, through the point P. The directional derivatives are the generalization of the partial derivatives; where in one of the coordinate axis is always parallel in direction.

Mathematical Definition of Directional Derivatives
The directional derivative is defined as the rate of change of the scalar function f(x) = f(x1, x2, x3... xn) along the unit vector u = (u1, u2, u3….. un), in which the unit vector is defined by a limit in terms of h. Therefore the mathematical definition for the directional derivative can be written as,

 Du f(x) = lim h->0+ f(x+hu) – f(x) / h.

Derivation of Directional derivative

It is simple method to find the directional derivative formula which will be used to simplify the problems. Let us define a function that is made up of a single variable, such as g(z) which equals f(sum of x0 + az and y0 + bz). Here x0, y0, a, b are some numbers which are fixed. Now, z is the letter which does not represent a fixed number.  Then, by the actual definition, for a single variable function we know that, g’(z) equals the limit which tends to h->0 of g (sum of z and h) – g(z) whole divided by h.  Now applying the value of the derivative z as zero, we will get, g’(0) which equals limit which tends to h->0 of subtraction of g (h) and g(0) whole divided by h. Substituting the equation of g(z) mentioned earlier we will get g’(0) as lim h->0 of f(x0 + az, y0 + bz) – f(x0,y0) whole divided by h. Then this equation will be obviously equal to the Du f(x0, y0). Therefore we will finally come with a relationship of g’(0) which equals Du f(x0, y0). Now we can rewrite g(z) as g(z) = f(x, y) where x will be equal to x0 + az and y will be equal to y0 + bz. Then, applying the chain rule we will get the equation as,  g’(z) = dg/dz = Sum of (multiplication of df/dx and dx/dz ) and (multiplication of df/dy and dy/dz), which will be equal to sum of fx (x,y) a and  fy (x,y).After taking z=0, we will get x as x0 and y as y0 and applying these, we will get, g’(0) as sum of            fx (x0,y0) a and fy (x0,y0) b. Now equating this equation with the g’(z) equation of very first we derived, we will arrive with a formula as shown below:

Du f(x, y) = fx (x, y) a + fy (x, y) b.

This equation will provide a simple way of calculation than the previous limit based definition.
Directional derivatives are used in solving the matrix symmetrical valued functions; especially the second directional derivative and it plays a vital role in solving non linear type problems.

Algebra Help

Algebra is the first branch of math which students come across, that is a complete change from arithmetic and basic geometry that students are used to studying. Algebra introduces the concept of the unknown variable and teaches students how the unknown can be derived from known parameters. Making use of both numbers and alphabets, algebra is a whole new ball game and needs complete attention right from the beginning .

How do keep up with the subject from the day you start learning it? To begin with, algebra is not as difficult as many students claim it is. Generally, students who require a little more time and explanation to understand algebraic concepts, seldom get it and as a result, come to the conclusion that it is hopelessly difficult. Needless to say, this leaves them with precious little motivation to continue learning the subject.

One of the first steps to learning algebra is to ensure that you understand each concept or equation when it is taught. Too many students tend to make light of their lack of clarity, telling themselves they'll take the time to clear it later on. Instead, take a proactive approach and ask a lot of questions in class. You don't have to worry about sounding dumb since it's a new subject and everyone is just as clueless as you are. Moreover when students ask questions, it helps teachers gauge the level of understanding among the students and proceed accordingly.

Memorize any equations or formulas immediately. You will be using them a lot so it's good to familiarize yourself with them and learn when and how to use these formulas. The best way to commit them to memory is to write them down. Practice sessions are very important so try to make time for them on a daily basis. Include different types of questions and don't just stick with the ones your teacher provides you with. As you get more comfortable with algebra and learn the formulas, vary the difficulty level to challenge yourself and learn new methods. It will also ensure that you are prepared for any question during the exams.

If you find that things are going above your head, then get help with algebra straightaway. Irrespective of whom you go to for help, they will find it easier to start from the beginning rather than having to finish the whole lot two weeks prior to the exam. For minor doubts you can approach your teacher or a classmate to point you in the right direction. For more extensive help, use a tutoring service which provides good quality, experienced tutors who will explain the theory, clear any doubts, and provide practice material. Algebra help online can be found on a number of websites which feature detailed tutorials as well as one on one tutoring, as per students' convenience.

Surface area of an ellipse cross section

Area of ellipse:
Like a circle or any other closed figure, an ellipse is also a closed figure. Therefore it is possible to find its area. Area of an ellipse can be found using the values of a and b. But what are these a and b? That can be understood using the general equation of an ellipse.

General equation of an ellipse:
[(x-h)/a]^2 + [(y-k)/b]^2, where a is the semi major axis and b is the semi minor axis. (h,k) is the centre of the ellipse.

Area of ellipse formula:
From the above general equation we know that the semi major axis is a and the semi minor axis is b. The formula for the area of an ellipse can now be written as follows:
Area = A =  pi * ab. The semi major axis can also be termed as the distance between the centre and the vertex, which is ‘a’ here. The semi minor axis can also be terms as the distance between the centre and the co-vertex, which is ‘b’ here.  See the picture below:

Implicit equation of an ellipse:

Another method for finding the area of an ellipse is using its implicit equation. The general form of the implicit equation of an ellipse is:

Px^2 + Qxy + Ry^2 = 1

To calculate area of an ellipse if the equation of the ellipse is given in this form, we use the following formula:

Area = A = 2 pi/√(4PR – Q^2)

Let us now try to understand how to calculate the area of an ellipse using an example.

Example 1: 
Find the area of an ellipse with centre at (2,3), length of major axis being 6 cm and length of minor axis being 4 cm.

Solution:
 In this problem, major axis = 6, therefore semi major axis = a = 6/2 = 3 cm
Minor axis  = 4, therefore semi minor axis = b = 4/2 = 2 cm
Using the formula for area of ellipse,
A =pi* ab =  pi * 3 * 2 = 6 pi cm^2

Statistics Mode

Mode definition (math):
In statistics, a branch of mathematics, mode of a set of observations (also called data) is the particular observation that occurs most number of times. In other words, the observation with the maximum frequency is the mode of the data.

Finding mode:
The following steps are to be followed when calculating the mode of a set of observations of ungrouped data.
Step 1: Arrange the data in ascending order
Step 2: Look for the entry that occurs the most number of times.
Step 3: The number found in the step 2 above is your mode.

Mode Math examples:
To properly understand what is a mode in math or statistics, let  us try the following examples:

Example 1: The height of 10 students in centimeters is as follows, find the mode of the data.
145, 142, 143, 146, 144, 146, 143, 141, 140, 143
Solution:
Step 1: Arrange the data in ascending order. So now the numbers would look like this:
140, 141, 142, 143, 143, 143, 144, 145, 146, 146
Step 2: Look for the entry that occurs the most number of times. In the above data, the number 143 occurs thrice, whereas all the other numbers occur either once or twice.
Step 3: The number found in step 2 is the mode. Therefore for this problem, mode = 143 since it occurs most number of times.

Example 2: The lengths of screws manufactured by a firm in inches are as follows:
1.55, 1.50, 1.52, 1.53, 1.50, 1.52, 1.51, 1.54, 1.56
Find the mode of the data.
Solution:
Step 1: Arranging the data in ascending order we have:
1.50, 1.50, 1.51, 1.52, 1.52, 1.53, 1.54, 1.55, 1.56
Step 2: From the above arrangement we see that two numbers 1.50 and 1.52, both occur twice.
Step 3: Therefore this data has two modes: 1.50 and 1.52. Such a data is called bimodal.

Example 3: 
The ages of students in college algebra class is as follows:
25, 23, 24, 22, 23, 25, 22, 21, 27, 28, 20, 29
Find the modes.
Solution:
Step 1: Arrange the data in ascending order
20, 21, 22, 22, 23, 23, 24, 25, 25, 27, 28, 29
Step 2: There are three numbers, each occurring twice. They are 22, 23 and 25.
Step 3: These three numbers are therefore the modes. Such a data is called trimodal.
If there are more than three modes in a data set, it is called multimodal data.

Identifying conic sections

See the picture above. Consider a double cone as shown in the picture. Also consider a plane. Let this plane intersect the double cone. The way in which the plane intersects the double cone, would give rise to conic sections. As we can see in the picture the axis of the cone is vertical.

1. Conic sections circles:  If the intersecting plane is perfectly horizontal, that means that if the intersecting plane is perpendicular to the vertical axis, then the conic section thus produced would be a circle.

2. Conic sections ellipse: If the intersecting plane is inclined to the vertical at an angle that is between α to 90 degrees, where α is the angle between the vertical axis and the slant length of the cone, then the conic section thus produced would be an ellipse.

3. Conic sections parabola: If the intersecting plane is inclined to the vertical axis at an angle that is between 0 to α (α is same as described above), then the conic section produced would be a parabola.

4. Conic sections hyperbola: If the intersecting plane is parallel to the vertical axis, then the conic section thus produced would be a hyperbola.

Identifying conic sections:
Conic sections can be identified by two methods. They are, graphical and algebraic.
Graphical method of identifying conic sections:
The graphs of various conic sections are as shown below:

1. Graph of a circle:

The graph of a circle would look like above, it would have a centre and the radius on all sides would be equal.

2.Graph of an ellipse:

The graph of an ellipse can look like any of the two above figures. One is a horizontal ellipse and other is a vertical ellipse. An ellipse would have a major and a minor axis, two vertices and two co-vertices.

3.Graph of a parabola:

A parabolic curve would look like above. It would essentially have a vertex, a focus, a directrix and an axis of symmetry.

A hyperbola can be horizontal or vertical. Algebraic method for identifying conic sections:
In a conic sections practice problem, the equation of the curve should be similar to any of the following general equations:

Equation of the curve	Type of conic section
(x-h)^2 + (y-k)^2 = r^2	Circle
((x-h)/a)^2 + ((y-k)/b)^2 = 1	Ellipse
y-k = 4a(x-h)^2	Parabola
((x-h)/a)^2 - ((y-k)/b)^2 = 1	Hyperbola

Introduction to discrete math applications

Discrete mathematics is defined as the study of mathematical structures that are logically discrete rather than continuous. Discrete mathematics had a concept in topics in "continuous mathematics" such as calculus and analysis

The set of conepts studied in discrete mathematics can be finite or infinite. Some of the mathematical relations often considered i part of discrete mathematics are Boolean algebra, the mathematics of social choice, linear programming, and number theory

Discrete mathematics includes sets, functions and relations, matrix algebra, combinatory and finite probability, graph theory, finite differences and recurrence relations, logic, mathematical induction, and algorithmic thinking.

Applications of Discrete Mathemetics:

It can be applicable in various fields such as combinational analysis, functional system theory, codings, crptology,etc

It can also be used in graph theory, probabilistic problems of discrete mathemetics, and it has the applications of algorithms and their complexity and computational problems of number theory and of algebra

It can be also used in vital, exciting and useful information about mathemetics that will be very useful to taught the lower and higher grade

Problems by Using Discrete Math Applications:

1. By using discrete mathematics find the value of n.

(1). (n+ 1)! = 10 * 9(n!)

Solution:

(n+1)! = 10 * 9(n!)

here expanding the factorization that gives

(n + 1) n! = 10 * 9n!

now cancel n! on both the sides

we get n+1 = 90

n = 90 – 1 = 89

therefore n = 90 .

(2). 1/4! + 1/5! = n/6!

Solution:

1/4 + 1/5(4!) = n/6.5(4!)

now cancel 4! On both sides

1 + 1/5 = n/6.5

6/5 = n/30

n = (30 * 6)/5 = 180 /5 = 36

Therfore we get n = 36.

2. Find the following factorial question by using discrete mathemetic

Where n = 4 and r = 2 find the values of

(1). n! / r!

Solution:

n! / r! = 4! / 2! = 4.3.2.1 / 2.1 = 12.

Therfore n! / r! =12

(2). n! /r!(n- r )!

Solution:

n! / r!(n-r)! = 4! / 2!(4-2 )!

= 4.3.2.1 / 2!

= 4.3.2.1 / 2.1

therefore n! / r!(n-r)! =12.

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.