What is Mathemathics: Definition and 104 Discussions
Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.
I'm interested in having a career in physics (I'm leaning more towards theoretical work instead of experimental aspect of the subject). I'm currently doing my undergraduate studies and decided to pursue Mathematics as a minor subject. These are the courses that I'm planning to take...
I proceeded as follows
$$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$
$$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$
$$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$
$$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$
Which leads...
I’ve attached my attempt. I’ve tried to use triangle inequality formula to attempt, but it seems I got the value which is larger than 1. Which step am I wrong? Also, it seems I cannot neglect the minus sign in front of e^(N+1/2)*2pi. How can I deal with that?
Here’s my work:
The integrating factor I find is (x^(2)-1)^1/2. The self adjoint form I find is
-d/dx (((1-x^(2))^(3/2))*dy/dx))=k(x^(2)-1)^(1/2).
Am I right?
Hi everyone, Im new to the forum and physics in general.
I recently started to dig a bit into popular science on quantum mechanics, cosmology, general relativity etc. and found it super interesting ( as everyone on this forum I assume ). I think the time has come for me to dig a bit deeper into...
Do you know about books and websites about manipulatives in mathematics teaching for secondary schools?
I think manipulatives are incredibly useful in teaching Mathematics.
Thank you for your support.
In order to write next step in all four equations above l used the definition of radicals. ##\sqrt a=b## means ##b^2=a##. Squaring both sides also works. I don’t know if it’s right. I mean I read that ##(\sqrt a)^2=a##. But I don’t know if we can apply this on expressions.
Main problem is if we...
Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus?
Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
I’m Cosby from Ghana.A student of the University of Education-Winneba in Ghana..
I’m in my first year and I’m trying to get help yo develop my mathematical skills in order to become a good mathematician.And that is why I joined this family.
Firstly I converted the given equation to a quadratic equation which is
##z^2- (\sqrt3)z+1=0##
I got two solutions:
1st sol ##z=\frac {(\sqrt3 + i)} {2}##
2nd sol ## z=\frac {(\sqrt3 -1)} {2}##
Then I found modulus and argument for both solution . Modulus=1 Arguments are ##\frac...
Just to remind, ##C_\ell## is the variance of random variables ##a_{\ell m}## following a Gaussian PDF (in spherical harmonics of Legendre) :
##C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)##
1)...
I'am interested in the notation of the approximation in quantum phase estimation algorithm.
In the literature there are different definitions, which I divide into two cases here. Both different in their definition of the ##\delta##. In both cases I start with a quote of the source and show an...
I am a 10(high school)+ 2 years student. I have a basic knowledge of calculus, trigonometry, algebra, arithmetic. I need to advance as an undergraduate to start my physics journey. I want a mathematics book for physics which if it will cover some basics in early chapters will be of much help...
Long story short, my career goals are to work as a researcher/research scientist in machine learning or to develop models/algorithms for forecasting such as in financial markets or otherwise. Fundamentally, I find think this is quite similar to physics as a science, but that is for another...
Do the stereotypical nerds and geeks really exist? I am talking about socially awkward timid loners who are also highly intelligent in Science, Mathematics, Technology, and Engineering.
What happens to them later in real life? Do they really end up as unemployed or menial workers because of...
Hello, I am a very experienced Mathematician with a BSc Honours degree in Mathematics and one year MSc studies in Operational Research in Sussex and London Universities respectively.
I am interested in Advanced Calculus, Algebras, Positivity in Algebraic Geometry, The standard Model, and many...
I am trying to calculate the interaction energy of two interpenetrating spheres of uniform charge density. Here is my work:
First I want to calculate the electric potential of one sphere as following;
$$\Phi(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int...
A system is linear if it satisfies the properties of superposition and homogeneity.
Superposition: adding the inputs of two systems results in the addition of the two outputs.
Ex) x1(t) + x2(t) = y1(t) + y2(t)
Homogeneity: multiplying the input by some scalar value is equal to the output...
I just finished my freshman year in computer engineering at my college, and after some physics and materials science classes I noticed a pattern. Everybody—at least in my class, I don't about the rest of the world—is just memorizing formulas, myself included. I'm really trying to not just...
As I have said before, I am in calculus class for the first time. I am doing really well in the class, however because of how my mind works, I’m always asking questions to know more, even when it’s too advanced for me. I just like to ponder and think about “what if” I know it’s probably not good...
Hi everyone , Math Doubts is joined in Physics Forums today to interact with all of you. We would like to help you in solving your math problems and clearing your doubts, and also wants to interact with mathematicians and scientists for knowledge sharing.
I am attempting to find the sine representation of cos 2x by letting
$$f(x) = \cos2x, x>0$$ and $$-\cos2x, x<0$$
Which is an odd function. Hence using $$b_n = \dfrac{2}{l} \int^\pi _0 f(x) \sin(\dfrac{n\pi x}{l})dx$$ I obtain $$b_n = \dfrac{2n}{\pi} \left( \dfrac{(-1)^n - 1}{4-n^2} \right)$$...
Hello I hope someone can help me, as i am kinda stuck for the moment.
As you can see, the assignment states that I need to find the poles from the closed loop transfer function.
I plan on doing so, by using block diagram reduction method.
This is as fare as I've come, and can't come further...
Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2
starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2=
[(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2
For the derivative in respect to y...
Homework Statement
I am carrying out a regression for diameter of a part
Homework Equations
Diameter = -0.0531052 + 0.0443237 * exp (-0.0103633 * 'Time elapsed')
if diameter is -0.052
then can some one please calculate the value for time elapsed
would you please explain the steps
The...
I'm in trouble trying to understand the expression ##t= \frac{1}{\omega} cos^{-1}(x/A)## that comes from ##x = Acos(\omega t)##, in which ##A## is the amplitude, ##t## is time and ##x## is displacement.
When ##x = 0##, ##t = \frac{\pi}{2\omega} ##, shouldn't it be 0 since there was no movement?
Homework Statement
Given:
Ψ and Φ are orthonormal find
(Ψ + Φ)^2
Homework Equations
None
The Attempt at a Solution
Since they are orthonormal functions then can i do this?
(Ψ + Φ) = (Ψ + Φ)(Ψ* + Φ*)?
I’m using an inductive heater to try to melt some 99.85% polycrystalline silicon, as would occur in the Czochralski process, but the silicon workpiece is not melting let alone even getting hot. The silicon workpiece is about the size of a playing dice. It becomes conductive at 200 F and becomes...
Hello every one,
My name is Stenli Suryadinata from Indonesia.
I found that this site was a great place to learn and share.
I wish to learn and share interesting things with all of you
Thank you
I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll.
In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution ##\bar{x}## is large , values of ##P(x)## for adjacent values of x are not greatly different from each...
Hi there , well my questions is in the title but i have to say some things :
I have multiples mathematiques and physics courses calc-based and so i want to review the basic .
I already search around this forum and others and i find some books who looks popular here there are :
- Pre-Calculus...
It seems like we keep chasing "reality", and by "reality" I think Physicists would mean the apparent rules of quantum physics which we hope would (if applied) lead to all the apparent known rules of macro-physics.
However ... It seems like we have had to create a few things to do that:
1. Ideas...
The asks for us to find the nature of roots of the following equation ,i.e,rational or irrational nature of the roots:
the Equation is : http://www5a.wolframalpha.com/Calculate/MSP/MSP7106108efdacf0gb84ie000039f29d33b6hie7ic?MSPStoreType=image/gif&s=56&w=63.&h=18.x^5+x=5
I have been able to...
Hello! New here. Sort of. I have a major in math. I'd like to specialize in data science for my master's degree. This is my 2nd account here. I don't know how to delete the other one. Can anyone tell? Thanks.
Hello Everyone,
So I have been looking for some time now to find an online program in EE/mathematics/physics. There are a number of programs I have found (purdue, johns hopkins, etc) but the prices are absolutly staggering. I have paid my dues to the student loan machine with my BS degree in...
Hi.
Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build.
Category Theory is at the highest level of abstraction; then...
Hello.
I’d like to self-learn physics. basically starting from a zero knowledge background to keep things simple.
I have asked before and some individuals said that it is using your time to learn it once you know calculus? Is this true or should I start learning the basics now so that when I...
Homework Statement
hello, I'm currently studying simpsons rule (unrelated) however the method requires the answer to cos^2(1^2) the answer given by my tutor is 0.2919, I have been unable to get this answer after inputting cos in various ways I always get 0.9997, which is right and if 0.2919 is...
Hi, currently I am having a dilemma trying to decide whether I want to do a 4 year Msci in physics or mathematics.
I am about to finish a degree in computer science I had started when I was 13 or 14 with the open university. Although, I still want to go to a real university for mathematics or...
Homework Statement
Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.
Homework Equations
f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)
The Attempt at a Solution
I...