# 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. ### I Multivariable fundamental calculus theorem in Wald

i want to prove that if ##F:\mathbb{R}^n\to\mathbb{R}## is a differentiable function, then $$F(x)=F(a)+\sum_{i=1}^n(x^i-a^i)H_i(x)$$ where ##H_i(a)=\frac{\partial F}{\partial x^i}\bigg|_{x=a}##. the hint is that with the 1-dimensional case, convert the integral into one with limits from ##0## to...
2. ### Differentiate the given integral

My take: $$\int_{x^2}^{2x} \sin t \, dt$$ using the fundamental theorem of calculus we shall have, $$\int_{x^2}^{2x} \sin t \, dt=-2x \sin x^2 +2 \sin 2x$$ I also wanted to check my answer, i did this by, $$\int [-2x \sin x^2 +2 \sin 2x] dx$$ for the integration of the first part i.e...
3. ### I Differential operator in multivariable fundamental theorem

I'm referring to this result: But I'm not sure what happens if I apply a linear differential operator to both sides (like a derivation ##D##) - more specifically I'm not sure at what point should each term be evaluated. Acting ##D## on both sides I'll get...
4. ### Find the largest interval on which f is increasing

Attached here is the full problem I am doing. I went through the problem and got my final answer which I thought was correct. Here is my work. They tell me I am wrong. Not sure where is the mistake.
5. ### B Extending the Fundamental Theorem of Arithmetic to the rationals

The Fundamental Theorem of Arithmetic essentially states that any positive whole number n can be written as: ##n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \cdot \dots## where ##p_1##, ##p_2##, ##p_3##, etc. are all the primes, and ##a_1##, ##a_2##, ##a_3##, etc. are non-negative integers...
6. ### I Decomposition per the Fundamental Theorem of Finite Abelian Groups

According to the book I am using, one can decompose a finite abelian group uniquely as a direct sum of cyclic groups with prime power orders. Uniquely meaning that the structures in the group somehow force you to one particular decomposition for any given group. Unfortunately, the book gives no...
7. ### I The Fundamental Theorem of Arithmetic and Rational Numbers

The fundamental theorem of arithmetic applies to prime factorizations of whole numbers. Can this theorem also correctly be invoked for all rational numbers? For example, if we take the number 3.25, it can be expressed as 13/4. This can be expressed as 13/2 x 1/2. This cannot be broken...
8. ### I Fundamental Theorem of Algebra: Proof

I found this video showing an elementary proof of the FTA.
9. ### B A proof of the fundamental theorem of calculus

is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it? i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it. i know...
10. ### I Can Fundamental Theorem Algebra be proven by simple DOF?

A monic polynomial of degree N has N number of coefficients. The product of N number of linear factors has N number of free terms. A complex number has 2 DOF. Therefore, both a monic polynomial and the product of free terms have 2N number of DOF of real values. Thus, it must be possible to...
11. ### I On the topological proof of the Fundamental Theorem of Algebra

Sorry for the misspelling, but this forum doesn't allow enough characters for the title. The title should be: For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots? I...
12. ### Insights A Numerical Insight for the Fundamental Theorem of Calculus - Comments

Greg Bernhardt submitted a new blog post A Numerical Insight for the Fundamental Theorem of Calculus Continue reading the Original Blog Post.
13. ### Application of the Fundamental Theorem of Calculus (cosmological red-shift)

Homework Statement [/B] I am stuck on the section of my lecture notes attached, where it says that equation 4.20 follows from 4.18 via an application of the fundamental theorem of calculus Homework Equations FoC: if ## f## is cts on ##[a,b]## then the function ...
14. ### MHB Fundamental theorem of calculus and more....

So as always I come here to make sure my maths homework is right and ask few questions to make sure I understand the topic. Here is my homework: Q.1 I'm fairly certain that this is correct, however, please check if I didn't do any stupid mistakes. Q.2 Same as above. Q.3 Now here is where the...
15. ### Vector Integration: Fundamental theorem use

Homework Statement Could someone illustrate why $$\int_{V} \nabla \cdot (f\vec{A}) \ dv = \int_{V} f( \nabla \cdot \vec{A} ) \ dv + \int_{V} \vec{A} \cdot (\nabla f ) \ dv = \oint f\vec{A} \cdot \ d\vec{a}$$ ? Homework EquationsThe Attempt at a Solution I understand that the integrand can...
16. ### A Fundamental Theorem of Quantum Measurements

The Fundamental Theorem of Quantum Measurements (see page 25 of these PDF notes) is given as follows: Every set of operators ##\{A_n \}_n## where ##n=1,...,N## that satisfies ##\sum_{n}A_{n}A^{\dagger}_{n} = I##, describes a possible measurement on a quantum system, where the measurement has...
17. ### MHB Another Question On B&S, Theorem 7.3.5 - Fundamental Theorem Of Calculus ...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 7: The Riemann Integral ... I need help in fully understanding yet another aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as...
18. ### MHB Fundamental Theorem Of Calculus (Second Form) - B&S Theorem 7.3.5 .... ....

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 7: The Riemann Integral ... I need help in fully understanding an aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as follows: In...
19. ### B Some help understanding integrals and calculus in general

So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...
20. ### I Question about the Fundamental Theorem of Algebra

Hi All, According to the fundamental theorem of algebra: "every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots". My question is: what about polynomials with degree say 2.3 or 3.02, as in the polynomial: ## p(x) =...
21. ### Proving the second fundamental theorem of calculus?

Homework Statement Show that Dx∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x). Homework Equations None The Attempt at a Solution I have ... = Dx(F(u)+C) = Dx(F(x-a)+C) = dxF(x) - dxF(a) = f(x)-f(a). My problem is that it should be...
22. ### I Visual interpretation of Fundamental Theorem of Calculus

Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If...
23. ### Calculating the definite integral using FTC pt 2

Homework Statement Sorry that I am not up on latex yet, but will describe the problem the best I can. On the interval of a=1 to b= 4 for X. ∫√5/√x. Homework EquationsThe Attempt at a Solution My text indicates the answer is 2√5. I have taken my anti derivative and plugged in b and subtracted...
24. ### Fundamental Theorem of Calc Problem using Chain Rule

Homework Statement F(x) = (integral from 1 to x^3) (t^2 - 10)/(t + 1) dt Evaluate F'(x) Homework Equations Using the chain rule The Attempt at a Solution Let u = x^3 Then: [((x^3)^2 - 10) / (x^3 + 1)] ⋅ 3x^2 *step cancelling powers of x from fraction* = (x^3 - 10)(3x^2) = 3x^5 - 30x^2 I am...
25. ### MHB Fundamental Theorem of Arithmetic - Bhattacharya et al - Ch. 2, Section 1

I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ... I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...
26. ### I Fundamental Theorem of Arithmetic - Bhattacharya et al

I am reading the book, Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul ... ... and am currently focused on Chapter 2: Integers, Real Numbers and Complex Numbers ...I need help with an aspect of the proof of the Fundamental Theorem of Arithmetic in Section 1.3 ... ...The...
27. ### A Trying an alternate Proof of the Fundamental Theorem

The proofs of the Fundamental Theorem of Calculus in the textbook I'm reading and those that I have found online, basically show us: 1) That when we apply the definition of the derivative to the integral of f (say F) below, we get f back. F(x) = \int_a^x f(t) dt 2) That any definite integral...
28. ### How to apply the fundamental theorem to partial derivatives?

Homework Statement "Under mild continuity restrictions, it is true that if ##F(x)=\int_a^b g(t,x)dt##, then ##F'(x)=\int_a^b g_x(t,x)dt##. Using this fact and the Chain Rule, we can find the derivative of ##F(x)=\int_{a}^{f(x)} g(t,x)dt## by letting ##G(u,x)=\int_a^u g(t,x)dt##, where...
29. ### Fundamental Theorem of Calculus: Part One

I am a little confused over part 1 of the fundamental theorem of calculus. Part 2 makes perfect sense to me. I guess my confusion is if we have an integral g(x) defined from [a, b], and we are looking at point x, how do we know that g'(x) = f(x)? It makes sense in the idea that they are...
30. ### The Fundamental Theorem of Calculus

Homework Statement I'm having trouble wrapping my head around this concept. I understand integration and differentiation individually. I even understand the algebraic manipulations that reveals their close relationship. However, the typical geometric interpretation of a 1-D curve being the...
31. ### Fundamental theorem of algebra and factoring?

Is the fundamental theorem of algebra (for polynomials on the complex plane) equivalent to the statement that any polynomial p of degree n>0 can be written p(z) = c(z - a_1 ) (z- a_2) \cdot \cdot \cdot (z - a_n ) or am I missing some subtle distinction? And if not equivalent, does the theorem...
32. ### Antiderivatives and the fundamental theorem

I know that according to the first fundamental theorem of calculus: $$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$ I also know that if ##F## is an antiderivative of ##f##, then the most general antiderivative is obtained by adding a constant. My question is, can every single antiderivative of ##f## be...
33. ### Fundamental theorem of calculus

"If ##f## is continuous on ##[a,b]## and: $$g(x) = \int_a^x f(t) dt$$ Then ##g## is continuous on ##[a,b]##, differentiable on ##(a,b)##, and ##g'(x) = f(x)##." This is the first fundamental theorem of calculus. I'm curious as to why ##g## is only differentiable on ##(a,b)##, but not ##[a,b]##.
34. ### Fundamental theorem of calculus for double integral

I was reading about double integral when a doubt came to my mind: how to find the antiderivative of the function f(x,y), like bellow, and compute the fundamental theorem of calculus for double integral? \int_{2}^{8} \int_{2}^{6} f(x,y) dx \wedge dy = ? OBS: It's not an exercise. I know how...
35. ### Fundamental Theorem of Calculus Problem

1. If g(x) = ∫ f(t) dt = xln x, find f(1) The ∫ has x^2 on top and 0 on bottom. 2. g'(x) = f(x) <--FTC1 The Attempt at a Solution g'(x) = f(x) u=x^2 g'(x) = u*lnu * 2x(derivative of inner function) g'(x) = 2x(x^2)ln(x^2) f(1) = 2(1)(1^2)ln(1^2) f(1) = 0, since ln(1) = 0 I...
36. ### Line Integral Example - mistake or am I missing something?

This is an example at the beginning of the section on the Fundamental Theorem for Line Integrals. 1. Homework Statement Find the work done by the gravitational field \vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x} in moving a particle from the point (3,4,12) to (2,2,0) along a piece wise...
37. ### Fundamental Theorem of Field extensions

Suppose F is a field and that ## f(x) ## is a non-constant polynomial in ##F[x]##. Since ##F[x] ## is a unique factorization domain, ## f(x) ## has an irreducible factor, ## p(x) ##. Then the fundamental theorem of field theory says that the field ## E = F[x]/<p(x)> ## contains a zero of ## f(x)...
38. ### Homework Question - Fundamental Theorem of Calc Example

Homework Statement Hi, I've been working through a practice problem for which I used the fundamental theorem of calculus, or one of its corollaries. The setup is a population changing over time. The population, P(t) at t = 0 is 6 billion. The limiting population as t goes to infinity is given...
39. ### The first fundamental theorem of calculus

Say I have the statement ##\int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y##. How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as ##F(x) = \int_{a}^{x}f(t)dt##. How...
40. ### MHB Fundamental theorem and limit proofs

Prove that the limit as n approaches infinity of ((2^n * n!)/n^n) equals to zero. The hint is to use Stirling's approximation. What is this?
41. ### Fundamental theorem of calculus - question & proof verifying

I understand that the fundamental theorem of calculus is essentially the statement that the derivative of the anti-derivative F evaluated at x\in (a,b) is equal to the value of the primitive function (integrand) f evaluated at x\in (a,b), i.e. F'(x)=f(x). However, can one imply directly from...
42. ### Fundamental Theorem of Calculus

In the fundamental theorem of calculus, why does f(x) have to be continuous in [a,b] for F(x) = \int_a^x f(x) dx ?
43. ### Proving the fundamental theorem of calculus using limits

Would it be a legitimate (valid) proof to use an \epsilon-\delta limit approach to prove the fundamental theorem of calculus? i.e. as the FTC states that if f is a continuous function on [a,b], then we can define a function F: [a,b]\rightarrow\mathbb{R} such that F(x)=\int_{a}^{x}f(t)dt Then F...
44. ### MHB What is Stoll's definition of the natural logarithm function?

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's definition of the natural logarithm function (page 234 -235) The relevant section of Stoll reads as follows: In this section we read: " ... ... To prove (a), consider the function L(ax), x \gt 0. By...
45. ### MHB Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of The Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2 Stoll's statement of Theorem 6.3.2 and its proof reads as follows: In the above proof we read: Since \mathscr{L}( \mathscr{P} ...
46. ### The Fundamental Theorem of Calculus

Homework Statement Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. Integral from [1 to 2] of (3/x^2 - 1) Homework Equations The answer is 1/2 f(x)dx= F(b) - F(a) The Attempt at a Solution I tried taking it to make it -x^-3 - 1x as the...
47. ### Solve for y(x) using the Fundamental Theorem of Calculus

Homework Statement Solve the integral equation for y(x): y(x) = 1 + ∫ { [y(t)]^2 / (1 + t^2) } dt (integral from 0 to x) See attached image for the equation in a nicer format. Homework Equations Fundamental Theorem of Calculus The Attempt at a Solution dy/dx = y(x)^2 / (1 + x^2) ∫ dy/y^2 = ∫...
48. ### Understanding the fundamental theorem of algebra

Dear all, I am trying to understand the fundamental theorem of algebra from the text (Alan F. Beardon, Algebra and Geometry) attached in this post. I have understood till the first two attachments and my question is from the 3rd attachment onwards. I will briefly describe what...
49. ### Why is the fundamental theorem of arithmetic special?

Why is it significant enough to be fundamental? Some people say that it is fundamental because it establishes the importance of primes as the building blocks of positive integers, but I could just as easily 'build up' the positive integers just by simply iterating +1's starting from 0.
50. ### MHB Fundamental theorem of galois theory

In fundamental theorem of galois theory,(statement): given that K be galois extension of F, G(K/F) be its galois group, S(K) be the set of all subfields of K containing F & S(G) be the set of all subgroups of G(K/F), mapping g: from S(K) to S(G) defined by g(H)=G(K/H), mapping h: from S(G) to...