bimbo36

i have been following maths like some sort of self learning process ...

i was able to improve a lot of things myself ...

i was able to understand some basic concepts a little bit more ...

i was able to narrow it down like this ...

 

mathematical expressions
equations in one variable
equations in two variables
system of 2 equations containing 2 variables
functions in one variable
functions in two variables

differential equations

first order differential equations
second order differential equations
higher order differential equations ...

linear differential equations
separable differential equations
exact differential equations
homogeneous differential equations
non homogeneous differential equations
using the method of undetermined coefficients ...



partial differential equations ...

then i also remember learning a lot of things about linear and non linear equations ...

http://www.mathsisfun.com/algebra/systems-linear-equations.html

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

http://www.sosmath.com/soe/SE211105/SE211105.html

 

A linear equation is always a polynomial of degree 1 (for example x+2y+3=0). In the two dimensional case, they always form lines; in other dimensions, they might also form planes, points, or hyperplanes. Their "shape" is always perfectly straight, with no curves of any kind. This is why we call them linear equations.

Every other equation is nonlinear. Higher degree polynomials are nonlinear. Trigonometric functions (like sin or cos) are nonlinear. Square roots are nonlinear. The main exception is if the nonlinear piece can evaluate to a constant--for example, sqrt(4)*x is linear because sqrt(4) is just 2, and 2x is linear.

Linear equations have some useful properties, mostly in that they are very easy to manipulate and solve. Although they are quite limited in what they can represent, it is often useful to try and approximate complicated systems using linear equations so that they will be easier to think about and deal with.

Nonlinear equations, for the most part, are much harder to solve and manipulate. Sometimes you need them--nature doesn't always work in straight lines, and nor do mathematicians--but generally speaking, you can only solve nonlinear equations if the systems are fairly small and simple. Solving a linear system with a million interacting variables is very doable with a computer, and most nonlinear solvers aren't going to get even close to that

 

for example , for a polynomial ...

a solution of a polynomial equation is also called a root of the polynomial ...

a value for the variable that makes the polynomial zero

if you can't find an exact expression, then you can use numerical methods to get approximations ...

with numerical methods you can choose how close to zero you want, and it will give you a value that's at least that close ...

let me add this definition too ...

so that it looks nice in one thread ...

The (standard) calculus is broken into two pieces.

i) Differential calculus - which is looking at the instantaneous rates of change of objects with respect to some variables. We have the notion of the derivative of a function.
ii) Integral calculus - which is calculating the area under curves, calculating volumes and so on. This is all given in terms if the (indefinite or definite) integral of a function.

The two notions are tied together via the fundamental theorem of calculus. This says that the derivative and indefinite integral are basically mutual inverses (but not quite).

 

I hope that gives you something to start with.

after all these , i was trying to learn differential equations ...

i was able to make some notes ... with the help of some pictures ...

can i please keep it , it was fun to read ...

which category of differential equation does it belong to ??

 

first order differential equations
second order differential equations
higher order differential equations ...

linear differential equations
separable differential equations
exact differential equations
homogeneous differential equations
non homogeneous differential equations
using the method of undetermined coefficients ...



partial differential equations ...

how do i solve this differential equation ??

ok , no problems ...

anyway those pictures sort of helped me a little bit ... from moving away from usually dull looking triangles... to understanding radian measurements ... till , to a point where i could ask questions about graphing trigonometric functions ...

and to how the repetitive cycles of some trigonometric functions are achieved ...

even though that last part isn't exactly that clear to me ...

then i googled something and found this link ... looks interesting ...

How to Graph a Sine Function ?

http://www.dummies.com/how-to/content/how-to-graph-a-sine-function.html

 

Knowing how to graph trig functions allows you to measure the movement of objects that move back and forth or up and down in a regular interval, such as pendulums. Sine functions are perfect ways of expressing this type of movement, because their graphs are repetitive and they oscillate (like a wave)

f(x) = sin x

It repeats itself every 2-pi radians ...

 

This repetition occurs because 2-pi radians is one trip around the unit circle — called the period of the sine graph — and after that, you start to go around again. Usually, you're asked to draw the graph to show one period of the function, because in this period you capture all possible values for sine before it starts repeating over and over again. The graph of sine is called periodic because of this repeating pattern

i really need a better understanding of this part ...

first of all , i am sorry for a bit of inappropriate content in it ... i was a bit excited because i could ask few proper questions from it ... i also had to arrange it that way quickly before few points escaped from my mind ...

please bear with the noobish attempt , this is an attempt for a deeper understanding .. before i get completely disconnected from the trigonometry ...

i would also appreciate it if i could keep this picture here , for some future reference .. even though it looks a bit messy ...

i am sorry for arranging pictures this way ... this was the only few pictures i could find ...


pictures removed


the second part of the questions would be like ...
pretend you know how to plot the graph of the trigonometric functions ...
then , when does this become a repetitive cycle ??

sorry about the pictures ... i sometimes i just arrange these things in my facebook profile ... so that i can ask better questions ...

 

The values in the quadrants show what result you get for the sine and cosine of various angles. The animation shows that if you take the cosine function and plot the values you get as you increase the angle, you end up with a sinusoidal wave

like a graphical approximate representation of a sinusoidal wave ??

 

This wave also occurs in many natural systems (a pendulum, a vibrating string, etc) as these are all examples of simple harmonic motion. And, as you noted originally, light also consists of sinusoidal electric and magnetic waves.

thanks , i am learning a lot of things ..

can i just post two more pictures associated with trigonometry ...? it looks interesting when i put together a few pics ... it looks sort of silly , but helps a noob like me when learning trigonometry ...

i also have few more doubts from the picture ...

i promise i wont post any more pictures associated with trigonometry after that ...

 

Trigonometric functions are also repetitive mathematical functions with a definite period between repetitions.

 

So it is not suprising that we often use combinations of trigonometric functions to represent waves mathematically.

 

We do not generate physical waves by running something through a trigonometric sausage maker.

i have few more doubts , and few more pics about the doubts ..

i dont really understand the change from the calculations in the quadrants to wave like output representation ??

Waves are repetitive physical phenomena that have a definite time interval between the repetition.

The time interval is called the period of the wave.

 

Trigonometric functions are also repetitive mathematical functions with a definite period between repetitions.

 

So it is not suprising that we often use combinations of trigonometric functions to represent waves mathematically.

 

We do not generate physical waves by running something through a trigonometric sausage maker.

 

So are you looking for actual examples of the use of the more complicated trigonometric formulae

 

or

 

are you looking to understand the generation of EM waves by electron acceleration?

thanks a lot for the nice explanations ...

the problem was that i am missing many sides of my trigonometry basics ... due to the not so simplicity of the trigonometry itself ...

at the same time i was trying to see some application side of the trigonometry too .... which was also a bit hard to see without the above picture i posted ...

 

are you looking to understand the generation of EM waves by electron acceleration?

yes i would really like a better understanding of this part where some good practical physics is involved ...

at the same time ...

 

So are you looking for actual examples of the use of the more complicated trigonometric formulae

this part too ...

i have been searching in google to understand some basics of it ...

and i have ended up with few basic question such as ...

what are some practical applications of sine, cosine , arc sine , etc ??

and

how does some trigonometric functions like ? y = sine(x) ?? ... f(x) = sine(x) ??? looks like .... ?

that would be like asking ... why the wave of the function sine(x) different from the wave of the function cose(x) ???

thanks for the reply .. i am sort of new to all these things ... this was mostly something i was trying to figure it out as part of improving some mathematics mostly trigonometry ,with the help of some applications to go with it ...

to be honest i could not find one proper application side ... to think about trigonometry on a bigger scale ...

and i was hoping something like this might help me see a bigger picture of trigonometry .... along with few of its applications ...

 

The acceleration imposed on electrons as they encounter magnetic fields produces electromagnetic waves. Their wavelength, frequency and intensity are controlled by a number of factors, including the size of the acceleration and how suddenly it is applied.

my question again is ..

the trigonometry you know , is it in the way the magnetic fields are input " somehow " to impose the acceleration on the electrons ? or is the trigonometry in the way the wave is output ??

first of all , once again i am sorry for posting this big picture .. please dont consider it as spam ...

??

first of all sorry for the delay ... i was bit busy with work ... i could not get a straightforward two hours in front of pc ...

thanks for explaining a lot of basic stuffs ...

simply depending on the books for answers like these is a bit boring ....

as for the last part ... i think its because ...

as the bigger x gets ... the bigger x ^2 gets ...

and for x=1/a or .... x =1/2 or , x=1/3 ... or x=1/4 ....

the value keeps getting smaller ....

 

As to whether the same function can give rise to more than one graph, the answer may be yes - that is why it is helpful to specify what sort of object the independent variable is (this is called the domain of our function), and equally what sort of object is the dependent variable (this is called the codomain)

this was very helpful ...

usually in questions i have seen , they don't mention much about the sort of object .. we are dealing with ?

wait .. i guess that's how some physics questions looks like ..?

but in some plain maths questions ... things like those are not usually mentioned that way ... its usually simply an x ...

maybe in physics type questions .. they are mentioned in a much more broader way .. like .. the point mass object x ...is something like ... a wave or light ? or something like that ???

thanks for the graphs and thanks for taking your valuable time to make a reply ...

i think i had some sort of simple physics equations type graphs in my head ... so that it was a bit of fun too to think about it ... mostly equations involving motions of objects at a specific point ... someone also said most of the physics type equations graph are dealt like point mass ...or something like that ...

so in the end .. i think its mostly about getting the numbers from the equation and plotting the graph like a point mass ... where the object x is only seen like a point mass ...

i haven't come across much graphs that looks like any thing else other than point mass ...

maybe i should look for more functions where the functions are capable of plotting regions ...

but probably the functions are then going to look a bit more complex .. and i might have a hard time understanding the functions themselves ...

its like most of the graphs in physics related problems are a bit advanced for me ... i am only thinking about some simple physics equations involving velocity .. acceleration ... etc ... where we could do something about the points in graph which represents something at a given point ...

i don't know how a function of a point mass wave looks like on a graph ...?

or what functions are used to represent the more advanced radio waves ... in graph .... ??

i am probably messing up my head too much ...

i am not sure if most of the physics type equations are deterministic in nature ....

anyway thanks ...

this probably looks like i have to be looking into a physics book rather than college level maths books ...

maybe i should simply stick with the basics ....

its supposed to be same .. yes ...

because the numbers you get from the same equation will always be the same ...

but how you draw those numbers is a completely different thing ???

first of all ... thanks for letting me keep the youtube video and picture .. i thought somebody was going to remove it and call it spam ...

thanks for not doing that ...

i am actually trying to improve my maths from some basics .. which is why my questions looks a bit dumb ...

i have few more basic questions to ask ...

i am trying to picture an expression or equation with which i can try to do something ...

take this one for example ...

i dont know what i could possibly do with a motion of an object in a graph ??

i have been trying to narrow it down to some simple things ...

i just cant come up with a proper question to ask ...

you can try to find its roots .. differentiate it , integrate it ... ???

you can try to see it like a point mass ...polytopes or NURBS ... ??

those are a bit of like new terms to me ...

please help with some answers ...

as i don't know where i am heading with all these doubts in my mind ...

i have few more dumb questions about some basic stuffs ...

is it always possible to plot the graphs of algebraic expressions and equations on the graph ???

i always had these doubts about variables ...how they were going to look like in a graph ?? about expressions , equations ... and how they are represented in the graph ...

are they only going to look like simply points on the graph ??? or you can call it a region ? or something like that ???

i have been trying to stare into books at free times... trying to fill the missing parts ... its a bit exhaustive .. sometimes i want to look into the books ... then i get double exhausted after staring into the books for a long time ...

is this something nice to follow .. ?? i am also trying to understand the concepts of variables involved in an equation.. and how they are going to look like in a graph ....

this is the only example i could come up with ....

thanks a lot .. these replies means a lot to me ... i thought i was never going to learn these things properly ... i have understood a lot of basic stuffs ... thanks to this forum and its members ....

now i have something to think about and pursue at free times ...

i have lot more to learn ...but now atleast i know where this is all heading ...

i was unable to get a bigger picture of this subject for a very long time ...

but after these discussions and replies .. i should be able to follow this a bit more deeply ...maybe not today or tommorow ... but someday its going to happen ....

i have few more questions ... i am almost exhausted trying to ask proper questions ... because its almost like i was trying to learn this from scratch ... it was a bit hard to find simple examples too ...

anyway the good thing is i am starting to understand this numerical methods types problems... i now sort of understand what others in this forum were trying to tell me about .. exact values and approximate values ..

i have few more doubts about some basic concepts ...

i think the one below all are the fixed point iteration type questions too...

 

fixed point iteration
the bisection method
the newton raphson method
the secant method

let me again start with .. with this simple example ...

i was trying to get a better picture with few images like these ...

it was helping me to take an initial guess ....

i think i figured it out that the initial guess ,(the initial root )... the answer of the question ... is like f(1.618) ... which makes the equation close to zero ... ??

this positive root is called a golden ratio ... right ??

is this like an answer to the question already ???

should i be looking for more values which makes the equation closer to zero ?

is that why we use numerical methods such as fixed point iteration ... ??

anyway then you put the values to fit the formulas of iteration methods ...

to get more approximate values which makes the equation closer to zero ???

thanks a lot for the answers ....

yea i almost forgot that , the value will be used to fit certain formulas ...

and the formulas should be able to give you more points in the graph after applying the appropriate numerical methods ...

then it becomes like possible points on that particular space .... ??? to do something ... ??

so this is what its mostly about .. f(1.618) being close to zero .. therefore a root of the equation ...

what practical value has this value has ???

i am trying to think if it has any practical application ... that value being there ???

is it like .. at f(1.618 ) .. the ball touches the ground ???

sorry for making it look like never ending dumb questions ...

do i keep typing that values in that plotter ??? anyway its giving me graph that looks like a straight forward line at different points?

and thats the approximate iterated answers ??

do i call it the possible roots of the equations ??

thanks ..

i was wondering how the results of this question ...

looks on the graph ???

so i i take this simple program and example here ??

the whole process is going to look like this ???

first of all i am sorry for using few pictures... because i am completly lost without it ...

i am also so lost and confused between all these equations ... the terminologies and stuffs ..

we use numerical methods depending on the type of equations ...

the equations could be linear or non linear ... right ?

depending on the type of equation ...

we use the numerical methods .. mentioned below ....

 

When we know the degree we can also give the polynomial a name:
Degree Name Example
0 Constant 7
1 Linear 4x+3
2 Quadratic x2−3x+2
3 Cubic 2x3−5x2
4 Quartic x4+3x−2

then we have simulaneous equations ...which looks like these ...

 

x+2y-3z=10
2x-3y-4z=1
y-3x+z=-8

these two numerical methods can be applied to it if you have to deal with equations like these ...

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/12-LinEqs_Direct.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/13-LinEqs_Indirect.pdf

now for non linear equations ...

these methods can be applied ....

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_2_Bisection.pdf

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_3_Newton.pdf

http://www.maths.dit.ie/~dmackey/lectures/Roots.pdf

these all are non linear, fixed point iteration type numerical methods ...

 

fixed point iteration
the bisection method
the newton raphson method
the secant method

is that an alright graph ?

so everytime i have to take an initial guess ? i have to plot its graph ...

i thought i almost had a grip on this subject ... now i lost it again ....

i am so lost ... why am i taking an initial guess for ? what is this initial guess consist of ???

i have this few doubts about taking an initial guess ... i am not sure how to do that when it comes to certain equations for solving them with numerical methods ...i dont know how to do that
you are supposed to take an initial guess when it comes to certain equations ...

is it about re arranging equations to fit certain formulas ... ???

do i have to take an initial guess of something in the equations when methods like these are involved ... ???

 

fixed point iteration
the bisection method
the newton raphson method
the secant method

few simple examples .. might look like this ...

that is fixed point iteration on a quadratic equation ...

they start by re arranging equations to apply the fixed point iteration ...

what is the initial guess there ??

and another example here ...

i am stuck with methods after methods for solving non linear equations ...

i think i sort of understand how equations are re arranged in fixed point iterations ...

are these methods somehow similiar ??

 

fixed point iteration
the bisection method
the newton raphson method
the secant method

http://www.maths.dit.ie/~dmackey/lectures/Roots.pdf

does these methods always involve re arranging equations ... to fit certain formulas ???

i am sorry for making all this look like spam ...

i am also sorry for deviating from the original topic ... but these all looks very similar ....

can i have a little bit of help ... with this fixed point iteration method ???

http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf

http://www.universityofcalicut.info/SDE/BSc_maths_numerical_methods.pdf

i am sorry if this looks like a mess ...

but fixed point iteration is the last method i am going to follow in numerical methods ..

some help or directions please before i completly quit this subject ... ???

but our questions in college were mostly oriented toward c program ...

it was called computer oriented numerical methods in C ...

the application of the methods were more important , rather than the quality of the questions ....

you mean those u v t are like x y z ... and therefore those are simultaneous equations... right ?

i learned few of those in high school physics class .. but dont remember it well now ... it must have been like 15 years ago ...

but after leaving college .. i am still following a subject which i failed badly ... the computer oriented numerical methods in c ....

because programming was hard .. mathematics was hard ... the combination of these two subjects were even harder ....