What is Fundamental theorem: Definition and 171 Discussions

In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself.

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  1. M

    Fundamental Theorem of Calculus Pt 2, multivariable integration?

    Homework Statement problem in attachment Homework Equations The Attempt at a Solution I can get f'(x) as sqrt(1 + (sinx)^2) and derive that to get the second derivative but as far as that I don't really get the concept behind this question will y be another function I have...
  2. K

    Find derivative using fundamental theorem of calculus part 1

    Homework Statement The Attempt at a Solution since cos(x^2) is on bottom i flipped it and so it becomes negative. then I got -[(1+cos(x^2))(-sin (x^2))(2x)] substituting with sqrt(pi/2) I keep getting the answer as sqrt(2pi) since the negatives cancel however it says the...
  3. D

    Proof of the Fundamental Theorem of Calculus

    It is proved in this topic (last answer): https://www.physicsforums.com/showthread.php?t=536987
  4. J

    How to state fundamental theorem of arithmetic in a formal way?

    I think a best informal way to state the theorem is Hardy's: every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes But clearly, this statement does not reveal the structure of the statement in the formal...
  5. A

    Fundamental Theorem of Space Curves

    This is not a question I need to work out but I'm trying to understand this theorem. My lecture notes state: 'This theorem states the existence of solutions to the Frenet - Serret Equations that, apart from the possibility of a rigid motion, are uniquely determined by any choice of smooth...
  6. F

    Trouble understanding the proof of fundamental theorem of algebra using

    *using complex analysis. (sorry couldn't post the entire thing in the thread title). It seems there is a common proof of the fundamental theorem of algebra using tools from complex analysis, namely Louisville's theorem and the max. modulus theorem. Here is an example proof from proof wiki...
  7. Saitama

    Fundamental theorem of counting

    Homework Statement How many natural numbers are there with the property that they can be expressed as the sum of the cubes of two natural numbers in two different ways. Homework Equations N/A The Attempt at a Solution I don't understand how should i start. :( Can somebody give...
  8. M

    Fundamental Theorem of Abelian Groups

    Show that there are two abelian groups of order 108 that have exactly one subgroup of order 3. 108 = 2^ 2 X 3 ^ 3 Using the fundamental theorem of finite abelian groups, we have Possible abelian groups of order 108 can be : Z108, Z4 + Z27, Z2+Z2+Z27, Z4+Z9+Z3, Z2+Z2+Z9+Z3, Z4+Z3+Z3+Z3...
  9. R

    Fundamental Theorem of Algebra Proof

    Hello, trying to figure out exactly what is going on in this question. Homework Statement (a) If P(z) is a nonconstant polynomial, show that |P(z)| > |P(0)| holds outside some disk R |z| ≤ R for some R > 0. Conclude that if the minimum value of |P(z)| for R z ≤ |R| occurs at z_o...
  10. J

    Fundamental Theorem of Calc, Part 1

    I'm confused. What's the difference between f(x) and f(t)?
  11. A

    Second Fundamental Theorem of Calculus Question

    Why do we say that f is a function of t and then take the derivative of the integral with respect to x? This confuses me because x is also the upper limit of integration.
  12. M

    Fundamental Theorem of Line Integrals

    If someone could link me to a tutorial on how to put in functions into a post, I would appreciate it, thanks. I am going to be putting in screen shots. Homework Statement http://img864.imageshack.us/img864/1517/scr1305133657.png" http://img864.imageshack.us/img864/1517/scr1305133657.png...
  13. J

    Fundamental Theorem Calc: Find f(4) from Integral

    Alright, so in my AP calc class we just got a worksheet and one of the questions i don't undersnat at allll! We have been learning about the Fundamental Theorem of Calculus recently, so I am guessing that is what this is about. Homework Statement Find f(4) if the integral (lower limit = 0 ...
  14. H

    Fundamental theorem of Orthogonality

    Hello there! I have a group represented by the following matricies: \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)\] ; 0.5\left( \begin{array}{cc} -1 & \sqrt{3} \\ -\sqrt{3} & -1 \end{array} \right)\] and\quad 0.5 \left( \begin{array}{cc} -1 & -\sqrt{3} \\ \sqrt{3} & -1...
  15. S

    Proving the Fundamental Theorem of Calculus Twice

    Homework Statement Complete the proof by using the Fundamental Theorem of Calculus TWICE to establish \int_c^d(\int_a^b f _{x}(x,y)dx)dy=...=\int_a^b(\int_{c}^{d}f_{x}(x,y) dy)dx Homework Equations I know that the FTC states that if g(x)=\int_a^x(f), then g'=f The Attempt at a...
  16. N

    Fundamental Theorem of Line Integral question

    The question is suppose that F is an inverse square force field, that is, F(r)=cr/|r^3| where c is some constant. r = xi + yj + zk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from those points to the origin...
  17. S

    The Fundamental Theorem of Algebra

    Hi all, I'm currently flicking through and old textbook and came across the following. "Every polynomial of the form , where has n linear factors over C...". What does it mean by http://latex.codecogs.com/gif.latex?a^{n}\neq0? Is this referring to some kind of complex index? This is all that...
  18. F

    Fundamental Theorem and Maxima

    Homework Statement Si(x) = \int_{0}^{x}{\frac{sin(t)}{t} dt At what values of x does this function have a local maximum? 2.Solutions [PLAIN]http://img833.imageshack.us/img833/701/27444263.png The Attempt at a Solution So I took the derivative and applying FTC and I got sin(x)/x = 0 sin(x)...
  19. A

    Who was the first to prove the fundamental theorem?

    Just curious who wrote the first proof of the fundamental theorem of calculus. Thanks.
  20. K

    Line Integral Fundamental Theorem

    Homework Statement Use Your Phi(from part 1) and the fundamental theorem of line integrals to evaluate the same line integral. (should get the same answer!) The Attempt at a Solution Phi from part 1: Phi = xy+ y^2 +C The line from before go from (0,2) to (-2,0) r(a) = (0,2)...
  21. S

    FTC fundamental theorem of calculus

    FTC "fundamental theorem of calculus Homework Statement Homework Equations FTC The Attempt at a Solution well i have not used FTC in a long time this is from my old lecture notes...how do i show that using FTC(the question show above)
  22. H

    Fundamental Theorem of Algebra Limit Proof

    Homework Statement prove that the limit of anxn + an-1xn-1 + ... + a1x1 + a0 as x goes to infinity equals infinity ** I forgot to mention that n is an odd number and this is for n>0 otherwise yes your counter example would be correct** Thanks for the quick responses by the way Otherwise, that...
  23. C

    The fundamental theorem of calculus

    Can we apply the fundamental theorem of calculus to an integrand that's a function of the differentiating variable? For example, can one still use the fundamental theorem of calculus when trying to find the derivative of this function: f(x) = \int_{0}^{x} \frac{cos(xt)}{t}dt
  24. J

    Fundamental Theorem of Arithmetic

    Homework Statement Theorem. (Fundamental Theorem of Arithmetic) Ever positive integer n has a prime factorization, which is unique except for reordering of the factors. Homework Equations 6.8 Definition. A prime factorization of n expresses n as a product of powers of distinct primes; the...
  25. D

    Role of mean value theorem in fundamental theorem of calculus proof

    Hi, I've been watching the MIT lectures on single variable calculus, and whilst proving FTC, he mentions that we since we know that: <$> f'(x) = g'(x) </$>, then by MVT we know that <$> f(x) = g(x) + C </$>. I have tried searching for somewhere where this implication is spelled out for me...
  26. T

    Question involving fundamental theorem of line integrals

    Homework Statement a) Use the fundamental theorem of line integrals to evaluate the line integral: ∫(2x/(x^2+y^2)^2)dx+(2y/(x^2+y^2)^2)dy (over C) Where C is the arc of the circle (x-4)^2+(y-5)^2=25 taken clockwise from (7,9) to (0,2). Explain why the fundamental theorem can be applied. b)...
  27. L

    Fundamental theorem of algebra

    I would like a reference for a purely algebraic proof of the fundamental theorem of algebra - or if you would like to supply a proof that would be even better.
  28. A

    Fundamental theorem of calculus in terms of Lebesgue integral

    What restrictions must we place on a real-valued function F for F(x) = \frac{d}{dx} \int_a^x F'(y)dy to hold, where "\int" is the Lebesgue integral?
  29. T

    Fundamental Theorem of Arithmetic

    Homework Statement Using the Fundamental Theorem of Arithmetic, prove that every positive integer can be written uniquely as a power of 2 and an odd number. Homework Equations The Attempt at a Solution Since the FTOA states that any integer can be written as a product of primes...
  30. N

    Fundamental theorem of calculus (something isn't right)

    The 2nd part of fundamental theorem of calculus says: Over what open interval does the formula F(x)=\int_{1}^{x}\frac{dt}{t} represent antiderivative of f(x)=1/x ? By looking at the theorem I would say that f(x) is continuous only for x \neq 0 So I would say that F(x) is defined on...
  31. L

    The Fundamental Theorem of Calculus, Part I

    According to Wikipedia, "the first fundamental theorem of calculus shows that an indefinite integration can be reversed by a differentiation." Am I wrong or is this theorem very simple? Indefinite integrals are the same as antiderivatives. So isn't this theorem simply stating that the...
  32. L

    Fundamental Theorem of Calculus and Line Integrals: Does it Apply?

    If I draw a random curve over a scalar field, then it is not generally true that the line integral of the scalar field over the curve equals the difference between the value of the antiderivatives of the scalar field at the beginning and finishing points of the curve, as one can clearly see by...
  33. M

    Fundamental theorem of calculus

    Homework Statement Does the function F(x)=int(sin(1/t)dt,0,x) (integral of sin(1/t) with lower limit 0 to upper limit x) have a derivative at x=0? Homework Equations The Attempt at a Solution I was thinking that F(x) shouldn't have a derivative at x=0 because the integrand isn't even...
  34. Phrak

    What is the Fundamental Theorem of Computer Science?

    What is the Fundamental Theorem of Computer Science? No such formally named theorem exists if I do a google search. But I'm very curious as to what students and professors of Computer Science might think it should be. Anyone?
  35. James889

    Question about the fundamental theorem of Calculus

    Hi, Suppose we're asked to find the derivative of the integral f(x)~=~\int_{-13}^{sin~x} \sqrt{1+t^2}~dt Now, the solution apparently looks like this: f(x)^{\prime} = \sqrt{1+sin^2(x)}\cdot~cos(x) Why? Why does the solution contain the upper limit `plugged in` ? A more sensible...
  36. W

    Fundamental Theorem of Calculus concept

    I just learned about the fundamental theorem of calculus. I can see that this ties together differentiation and intergration, but I was wondering what kind of problems can be solved by using this theorem? In other words, what can the theorem be applied to?
  37. A

    Vector calculus fundamental theorem corollaries

    Homework Statement Prove \int_{V}\nabla\ T d\tau\ = \oint_{S}Td\vec{a} Homework Equations Divergence theorem: \int_{V}(\nabla\bullet\vec{A})d\tau\ = \oint_{S}\vec{A}\bullet\ d\vec{a} The Attempt at a Solution By using the divergence theorem with the product rule for...
  38. F

    Fundamental Theorem of Arithmetic Problem

    Problem number 4 on the image has me stumped. I understand the problem (obviously not enough) and what its saying, I'm just having trouble putting it into a proof. Can i get a hint to get me started? Thanks http://img40.imageshack.us/i/asdasdjql.jpg/"
  39. J

    Fundamental Theorem of Calculus to find Derivative

    Use the Fundamental Theorem of Calculus to find the derivative of the function g(x) = \sqrt{x}\int sinx Ln(t) \frac{cos(t)}{t} dt g'(x) = lnx cosx / x. By integrating this function, you receive the function g(x). Then by differentiating g(x) you receive g'(x) which is what is...
  40. C

    Proving converse of fundamental theorem of cyclic groups

    Homework Statement If G is a finite abelian group that has one subgroup of order d for every divisor d of the order of G. Prove that G is cyclic. Homework Equations The Attempt at a Solution
  41. A

    Fundamental Theorem of Calculus properties

    Homework Statement Find a function f : [-1,1] ---> R such that f satisfies the following properties: a) f is continuous b) f is restricted to (-1,1) is differentiable c) its derivative f' is not differentiable on (-1,1) Homework Equations The Attempt at a Solution I kinda...
  42. P

    Fundamental Theorem of Calculus problem

    Homework Statement The FTC states that {d\over{dx}}\int_a^x f(t)dt = f(x) Now, how do I do something like {d\over{dx}}\int_a^{g(x)} f(t)dt = ?Homework Equations The Attempt at a Solution I know that it has to do with the chain rule, but I forgot my textbook at school and I can't seem to find it...
  43. R

    Fundamental theorem of calculus

    There are two theorems: The fundamental theorem of calculus: \int_{a}^{b}F'(x) = F(b) - F(a) And the theorem that states if f is continious on [a,b]and g:[a,b]->R is defined by g(x) = \int_{a}^{x}f(t) dt, then g is differentiable on (a,b) and; g'(x) = \frac{d}{dx}(\int^{x}_{a}f(t)) =...
  44. J

    Integral and Fundamental Theorem of Calculus

    Homework Statement Find 2 \int 0 \ (2-x^2)dx using def of the definite integral and using FTC Homework Equations The Attempt at a Solution any help on how to start would be great?
  45. I

    Find Local Max of f(x) w/ FTC2

    Homework Statement Question One: Find a continuous function f and a number a such that 2 + \int_{a}^{x} \frac {f(t)} {t^{6}} \,dt = 6 x^{-1} Question Two: At what value of x does the local max of f(x) occur? f(x) = \int_0^x \frac{ t^2 - 25 }{ 1+\cos^2(t)} dt The attempt at a solution I...
  46. S

    Fundamental theorem of calculus

    Homework Statement Evaluate each definite integral. ok I'm not sure how to do the integration sign, but... b=4, a=1 for (5yy^.5)+3y^.5)dy Homework Equations The Attempt at a Solution I'm not really sure what I'm doing wrong. i integrated...[tex]5y^2/2*y^(3/2)/(3/2)+3y^(3/2)/(3/2) then i...
  47. O

    What is the intuitive idea behind the fundamental theorem of calculus?

    Can somebody explain to me, geometrically and intuitively, the fundamental theorem of calculus? I understand that i can find the area between the graph of f'(x) and the x-axis where b>x>a by finding f(b)-f(a), but i don't understand why. Suppose something is too hard to integrate. Can i...
  48. H

    Ignoring Fundamental Theorem of Polynomials (a^n + b^n)

    Homework Statement Okay, basically why does (a_{}n + b_{}n) ignore the Fundamental Theory of Polynomials? Homework Equations ... I could post them here, but basically when n is odd (a_{n} + b_{n}) = a series that looks like this: (a+b) (a_{n-1} b_{0} - a_{n-2}b +a_{n-3}b_{2} + ... +...
  49. E

    Fundamental theorem of algebra

    [SOLVED] fundamental theorem of algebra Homework Statement Theorem: If F is a field, then every ideal of F[x] is principal. Use the above theorem to prove the equivalence between these two theorems: Fundamental Theorem of Algebra: Every nonconstant polynomial in C[x] has a zero in C...
  50. H

    Help w/First Fundamental Theorem of Calculus

    For the first fundamental theorem of calculus, I don't quite understand it... I think I got the integral part, but not the interval [a,x]... can you guys help me? thx
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