What is Mathematical: Definition and 1000 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 want to study GR from a mathematical point of view but I know almost no physics. Is this possible? And what textbook would be more geared towards this?
Also, what are the math prerequisites that I need? I have studied up to analysis on manifolds, some linear algebra and multi-linear...
I am no expert in formal logic, so please forgive me if this question sounds stupid. It's about a common pattern used in many mathematical proofs.
For me it' "obvious" or "trivial" - but I can't prove it.
For a friend of mine it's far from obvious or even wrong - but I don't get his point...
A ship has three cargo loads -forward, centre and after. The capacity limits are given:
Commodity Weight (in tonne) Volume (in cu. feet)
Forward 2000 100000
Centre 3000 135000
After 1500 30000
The following cargoes are offered. The ship owner may accept all or any part of each commodity...
Homework Statement
(5^2) + (11^2) + (17^2) +...+ (18n-1)^2
a)Write the sum in sigma notation
b)Using the sigma identities solve the sum (easy to do)Homework Equations
∑i = .5*k*(k+1) etc
The Attempt at a Solution
The problem I'm having is with the 25 and 121. I thought it was
∑ (6n-1)^2...
hi i have this homework question and I am not sure if my thought process is valid.
The Question:
let a, b and c be roots of the polynomial equation: x^3+px+q=0 and S(n)=(a^n)+(b^n)+(c^n)
now prove: that for S(n)= -p(S(n-2))-q(S(n-3)) for n>3my attempt:
-------------
first off...
Hi I have a problem about Inequality.
Let's suppose we have a inequality like this:
axb≥d and cxd≥d so I want to find connection between a and d its possible to do something like that
Thanks
Recall that the fibonacci sequence is defined as
{ f0=0; f1 = 1 and
{fn = f n - 1 + fn -2 for n 2
Prove by generalized mathematical induction that
fn = 1/sqrt(5)[ϕn - (-ϕ)-n]
where ϕ = [1+sqrt(5)]/2
is the golden ratio.. (This is known as de Moivre's formula.)
So...
I want to discuss about Chapter 8, equation 43 for any real number α(alpha)
The author tells "The continuity and monotonocity of E and L show that this definition
leads to the same result as the previously suggested one".(i.e.,Equ 33)
But I think properties of L is not required for the...
Hello, I'm a first year Physics student. I've decided to take on mathematical methods and be as thorough with it as I possibly can.
I am looking for books that cover vector calculus, Orthogonal transformations, tensors, curvilinear co-ordinates, index notations and the dirac-delta function in...
What are your favorite small numbers? Why?
Anyone can post in this thread, but the rules are:
1. The numbers must be single digit (you can cheat if you use hexadecimal, hint, hint).
2. 1 doesn't count (it's not prime, so go away).
3. If a closed form is not available, you can only use 20...
I am reading Principles Of Mathematical Analysis by Walter Rudin.
Chapter 8, Equation 28 needs some clarification.
We are taking a lim as h->0 on a series. Value of series is lim as n->INF on the squence of partial
sums. So first we take lim as n->INF on a sequence of partial sums and then...
I am not too used to reading mathematical text using symbols. I need some help confirming what I have in my textbook.
The book is describing a point in 3D space using the definition
R x R x R = {(x,y,z) | x,y,z \in R} and R is all real numbers
Using this source describing the...
Hi All,
Have the following given and would like some help with the 3 questions below. Any help will be greatly appreciated :)
City 1
TOTAL POPULATION CITY 1 100,000
Avg Age of Population 27.8
Median Age (Male) 20.5
Median Age (Female) 27.8
% Male Population 46.43%
% Female Population 53.57%...
Hi everybody. I am studying Biotechnology (B.Sc.) and we only have one course of Calculus and one of General Physics. I would like to continue my study with am M.Sc. in Computational Science or something similar so i need to study more math and physics (so first of all math!). I found the book...
OK, so the word "addiction" has obvious negative connotations, but am I alone - methinks not - in being far more than just "interested" in maths, and actually more of a border-line addict, always looking for that next discovery/fix...?
Bad mammal! lol (Heidy)(Heidy)(Heidy)
Homework Statement
A wave is represented by y = A sin (kx + ωt). Draw two cycles of the wave from x = 0 to x = 2λ at a) t = 0; b) t = T/4, where T = 1/f = 2∏/ω
Homework Equations
y = A sin (kx+ωt)
k = 2∏/λ (number of wave peaks)
The Attempt at a Solution
I had a really hard...
Hey!
I've just started to learn some mathematical induction and it's proving quite tricky. Hopefully someone could help me out a little with my questions!
I don't know how to format question on this forum but I'll try my best to accurately represent it:
n
sigma i * 2^i = (n-1) * (2^n+1) + 2...
I have a difficult problem on my research exploration for math. I am modeling the convection currents of boiling water in a pot, which is heated by an electric stove. There is a constant supply of energy at 8000 watts (the average for a stove) that heats the pot from the bottom of the pot...
I have a difficult problem on my research exploration for math. I am modeling the convection currents of boiling water in a pot, which is heated by an electric stove. There is a constant supply of energy at 8000 watts (the average for a stove) that heats the pot from the bottom of the pot...
In the attached equation from structural analysis, I don't understand the transition between the second and third equal signs. I think that u_{j,l}da_l = u_{j,1}da_1+u_{j,2}da_2+u_{j,3}da_3 , so doesn't equating da_j\delta_{lj} = da_l assume that l=j . can anyone help me see why this...
if the greater of two numbers is one more than twice the smaller and the smaller is x, what is the bigger? if the bigger is x, what is the smaller number?
my answer to the first part is 2x+1.
but i don't know how to represent the second condition in a mathematical form.
and can you also give...
1.
a.if john finished the job in 8 days, what was the part he finished in n days?
b. by how much 4k greater than 4h?
c. represent the amount of salt in x gallons of a 25% by volume of salt in solution water.
d. represent 3 consecutive integers if the smallest is n.
answers
a. n/8
b. 4h+4k
c. i...
Right now I'm a 2nd year biology major, focusing on neurobiology. During my first year I noticed that biology outside neuroscience and some physiology doesn't really interest me that much. So I started taking some maths and programming already during my 1st year. Now I'm taking mostly maths and...
Homework Statement
if A' = BxC/(A.BxC)
B' = CxA/(A.BxC)
C' = AxB/(A.BxC)
prove that A = B'xC'/(A'.B'xC')
B = C'xA'/(A'.B'xC')
C=(A'xB')/(A'.B'xC')Homework Equations
The Attempt at a Solution
I was sort of lost on how to do this. First I tried to simplify a', b', and c' and plug those into a...
Hi guys,
I studied mechanical engineering and I am currently a graduate student (turbulence modeling).
I am interested in the more formal/mathematical aspects of fluid dynamics, which are mainly worked in mathematical faculties and I would like to know if someone knows/have an idea about...
I'm a computer science major, computational biology in particular, that's looking to delve deeper into mathematics without actually double majoring. I've taken up to calculus 2, last topic covered was Taylor series. I've scoured numerous sources but to no avail. I've read that intro to logic and...
I'm working on a project on Logic and as part of it I want to write a selection of sentences in mathematical/logical syntax. I've consulted a mathemagician for the the first sentence but need confirmation on the other five, would someone be able to tell me if my maths lingo is correct?
What I...
I've always wondered do mathematical physicists tend to normally have a rigorous training in physics? I know (at least at most universities) mathematical physics is grouped under the math department rather than the physics department. So what training in physics do these individuals actually...
I've just started working through Rudin's Principles of Mathematical analysis. In my opinion, the choice of symbols doesn't promote "rigour" and encourages us to make needless assumptions.
For example, when defining an ordered field, the fourth axiom for addition states:
To every x...
With mathematical induction i should prove that is true for all natural numbers:http://img.tapatalk.com/d/13/10/18/u6epesu5.jpg
Im sorry beacuse i have inserted an image,but I am still not used to write it in [MATH] here...
Hello!
I am on my last year of study for my bachelor degree in Mechatronics. Looking back at my courses I can say that what I want to do with my life is mathematical modelling of systems (Be they technical, financial, whatever).
My favorite courses so far were the mathematical ones and the...
I want to address this post primarily to people who have already studied mathematical logic, or are currently studying it.
Since a while I've immersed myself in studying some math, and I must say that I started to enjoy pure mathematics. However, sometimes I don't really feel comfortable...
Not really sure where this question belongs, but it is based on a school assignment so I'll put it here.
I'm a Physics major and math minor, and I am about halfway through my first group of advanced physics courses (namely intro classes for analytical mechanics, modern physics, and theoretical...
Homework Statement
Not really sure where this question belongs, but it is based on a school assignment so I'll put it here.
I'm a Physics major and math minor, and I am about halfway through my first group of advanced physics courses (namely intro classes for analytical mechanics, modern...
I have been studying topology and abstract algebra for some years, and for a lot of time I have been having a hard time trying to understand the definition and concept of "preservation of mathematical structures".
For instance for binary operators a Homomorphism is said to preserve...
Hello everybody
Two questions concerning the notion of wave function.
1. Every quantum mechanics book describes the space of wave functions (or more generally of quantum states) of a given system as a Hilbert space. But correct me if I'm wrong QM also says that two elements of such a...
Hi, Physics Forums. I just have a quick question regarding which two math-type electives I should take as a physical chemistry major. Right now I am enrolled in Linear Algebra (Math 115A) and Mathematical Methods for Physicists (Physics 131) and plan on taking the second MM (Physics 132) next...
Hello^^ (I'm new here)
I want to know the mathematical tools i need to study continuum mechanics.
It would be great if someone give me a link that contains video lectures.
Thanks for help .
Is this anything to get excited about? I have no idea what it's about.
https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/
https://www.simonsfoundation.org/wp-content/uploads/2013/09/amplutihedron_span.jpg
Hello,
I was an electrical engineering student and knew how to utilize the necessary mathematical tools to solve anything that came my way.
I always had a deep latent predilection for mathematics but due to some constraints I was unable to spare much time on improving my knowledge of the subject...
Homework Statement
Let u be an arbitrary fixed unit vector and show that an vector b satisfies b^2 = (\vec{u} \cdot \vec{b}) + (\vec{u} \times \vec{b})^2 Explain this result in words, with the help of a picture.
Homework Equations
The Attempt at a Solution
I understand that the equations says...
Assuming a mathematical plane, does it have a top and a bottom or does defining them make the plane three dimensional?
Example: Given a flat, transparent plastic sheet. One draws a picture on it with a marker. If one turns the sheet over, in other words looking at the bottom of the sheet...