Gottfried Wilhelm (von) Leibniz (; German: [ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪbnɪts] or [ˈlaɪpnɪts]; 1 July 1646 [O.S. 21 June] – 14 November 1716) was a German philosopher, mathematician, scientist, diplomat and polymath. He is a prominent figure in both the history of philosophy and the history of mathematics. As a philosopher, he was one of the greatest representatives of 17th century rationalism. As a mathematician, his greatest achievement was the development of the main ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Mathematical works have consistently favored Leibniz's notation as the conventional expression of calculus. However, it was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found a consistent mathematical formulation by means of non-standard analysis. He was also a pioneer in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of nearly all digital (electronic, solid-state, discrete logic) computers, including the Von Neumann architecture, which is the standard design paradigm, or "computer architecture", followed from the second half of the 20th century, and into the 21st.
In philosophy and theology, Leibniz is most noted for his optimism, i.e. his conclusion that our world is, in a qualified sense, the best possible world that God could have created, a view that was often lampooned by other thinkers, such as Voltaire in his satirical novella Candide. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great early modern rationalists. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy also assimilates elements of the scholastic tradition, notably the assumption that some substantive knowledge of reality can be achieved by reasoning from first principles or prior definitions.
Leibniz made major contributions to physics and technology, and anticipated notions that surfaced much later in philosophy, probability theory, biology, medicine, geology, psychology, linguistics, and computer science. He wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibniz also contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and German, but also in English, Italian and Dutch. There is no complete gathering of the writings of Leibniz translated into English.
(This is from W. Greiner Quantum Mechanics, p. 293 from the topic of Ritz Variational Method)
1) Are ##\frac{\delta}{\delta \psi^{*}}## derivatives in equations 11.35a and 11.35b? If this is so, we can differentiate under the integral sign to get ##\int d^3x (\hat{H}\psi)## in equation 11.35a...
Electric potential at a point inside the charge distribution is:
##\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{V'-\delta}
\dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'##
where:
##\delta## is a small volume around point ##\mathbf{r}=\mathbf{r'}##
##\mathbf{r}##...
Homework Statement
Find the derivative of ##y=cos^3(πx)##
*Must be in Leibniz notation
Homework EquationsThe Attempt at a Solution
(i) $$Let~ w=y^3 , y=cos(u), u=πx$$
(ii) $$\frac{dw}{dy} = 3y^2,~ \frac{dy}{du} = -sin(u),~ \frac{du}{dx}=π$$
(iii) By the Chain Rule,
$$\frac{dw}{dx} =...
As part of the work I'm doing, I'm evaluating a contour integral:
$$\Omega \equiv \oint_{\Omega} \mathbf{f}(\mathbf{s}) \cdot \mathrm{d}\mathbf{s}$$
along the border of a region on a surface ##\mathbf{s}(u,v)##, where ##u,v## are local curvilinear coordinates, and where the surface itself is...
Homework Statement
Homework Equations
##V=V^u \partial_u ##
I am a bit confused with the notation used for the Lie Derivative of a vector field written as the commutator expression:
Not using the commutator expression I have:
## (L_vU)^u = V^u \partial_u U^v - U^u\partial_u V^v## (1)...
I know how to prove the quotient rule by using the definition of a derivative using limits (Newton's style). I just saw a proof of the product rule using Leibniz's concept of differentials on Wikipedia. https://en.wikipedia.org/wiki/Product_rule#Discovery
Does anyone know of a Leibniz-style...
I was wondering if I could get some pointers on how to at least start on this. In quantum mechanics we are using the WKB approximation, and we end up with a definite integral that looks like this:
∫(1 - a(cosh(x))-2)1/2 dx = ∫(1/cosh(x)) (1 - a(cosh(x))2)1/2 dx
where a is a positive constant...
Hello,
I have tried the integral below with Mathematica and it gives me the following solution:
##\frac{d}{dc}\int_{z^{-1}(c)}^{1} z(x)dx = -\frac{c}{z'(z^{-1}(c))}##
I am not quite sure where it gets it from...I think it can be separated and with differentiation the first part will be zero...
I want to evaluate
$$ \frac{d}{dt}\int_{0}^{^{\eta(t)}}\rho(p,t)dz $$
where p itself is $$ p=p(z,t) $$
I have the feeling I have to use Leibniz rule and/or chain rule, but I'm not sure how...
Thanks.
Homework Statement
1. How did they complete the square for these equations in the picture below? What was their thought process?
2. distance/velocity = time , velocity/acceleration = time , In leibniz notation how does this cancel out?
When you divide, how does it cancel out to give you a...
Homework Statement
Statement:[/B] Let y = u(x)v(x).
a) Find y' , y'', and y'''
b) The general formula for yn, the n-th derivative, is called Leibniz’ formula: it uses the same coefficients as the binomial theorem , and looks like https://i.gyazo.com/53728964c6b3ef142fd70f600c29e037.png
Use...
The entire first semester of my Calculus class we used Lagrange's notation, f'(x), f''(x), etc.. So at the beginning of second semester the teacher kinda casually switched over to Leibniz notation, dy/dx, which left all of the class dazed.
I understood it pretty well until she did a simple...
I was reading the Wikipedia page on Dynamism in order to get an idea of the motivation and thinking behind Liebniz's physics. In it there is this paragraph:
In the opening paragraph of Specimen dynamicum (1692), Leibniz begins by clarifying his intention to supersede the Cartesian account of...
Homework Statement
For x> 0 , let f(x) = $$\int _1^x \frac{log t dt }{1+t}$$
Then ## f(x) + f(1/x)## is equal to :
A. ##¼ (log x)^2 ##
B. ## ½ (log x)^2 ##
C. ##log x ##
D. ## ¼ log x^2 ##
Homework Equations
Suppose f(x) = $$\int _1^x g(t)$$
Then by Leibniz rule ,
f' (x) = g(x)
The...
Hi,
I am reading a paper that states: "We note that if an integrable function satisfies the Lipschitz condition of order one, then differentiation and integration can be interchanged. This provides a more compact way to take the derivative. Consequently, in our proofs, if an integrable function...
Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is...
Hi PF!
Can anyone help me with showing the following: $$\frac{\partial}{\partial x} \int_{f(x)}^{g(x)} L(x,y)dy = \int_{f(x)}^{g(x)} \frac{\partial L}{\partial x} dy + \frac{\partial g}{\partial x} L(g,y) - \frac{\partial f}{\partial x} L(f,y)$$
I understand this as the fundamental theorem if...
For a vector : ##\nabla_a V^b=\partial _a V^b+T^b_{ac}V^c##
I am trying to derive for a covector: ##\nabla_a w_b=\partial _a w_b+T^c_{ab}w_c##
I am told to use the Leibniz Rule and the definition that for a scalar ##f## : ##\nabla_a f =\partial_a f ## to do so
My thoughts:
Define ##w_b##...
Homework Statement
I have the functionu(x,t)=\frac{1}{2c}\int^{x+ct}_{x-ct}g(\xi)d\xiwhere g is continuously differentiable and c is a constant. I need to verify that this is a solution to the wave equation.
Homework Equations
My prof gave me the...
I never really understood leibniz notation. I know that dy/dx means differential of y with respect to x, but what do the 'd's mean? How come the second-order differential is d2y/dx2? What does that mean? And what does d/dx mean?
Homework Statement
Using Leibniz's rule to find dy/dx for
y = integral of e^-t from interval: [ln(x) to ln(x+1)]Homework Equations
The Attempt at a Solution
dy/dx = e^-(ln(x+1))*(1/(1+x))-e^-(ln(x))*(1/x)
= (x+1)/(1+x) - (x/x)
= 0
Im not sure what I'm doing wrong or how to properly use...
Remember the Newton's binomial theorem which says:
(x+y)^n = \sum_{r=0}^n {n \choose r} x^{n-r} y^r
where {n \choose r}=\frac{n!}{r! (n-r)!}
Now let's take a look at the Leibniz rule:
\frac{d^n}{dz^n}(xy)=\sum_{r=0}^n {n \choose r} \frac{d^{n-r} x}{dz^{n-r}} \frac{d^r y}{dz^r}...
Newton and Leibniz both had a method of differentiating. Newton had fluxions and Leibniz had something that resembles the modern derivative.
Historically, does anyone know how they went about calculating the derivative?
Homework Statement
Here is this problem:
I have the solution http://www.proofwiki.org/wiki/Leibniz%27s_Rule/One_Variable
This is where I get stuck..
Where it says: 'For the first summation, we separate the case k=n and then shift the indices up by 1.'
Why does this lead to the...
I had taken a multi-variable calculus course and since have misplaced my notes. I recall the prof inventing his own notation because somewhere partial derivatives using Leibniz notation don't show the correct path. I think it was something like if you had a function f(x,y)=z and y depended on x...
Hello,I am encountering some major confusion. When taking just garden variety f(x)=y derivatives of the form dy/dx, I don't encounter any problems. But recently I started taking derivatives of parametric equations, or switching things up using polar equations and I realized perhaps I'm not so...
Given f(x) = xe-x2 I can differentiate once and use Leibniz to show that for n greater than 1
f(n) = -2nf(n-2) - 2xf(n-1)
I want to show that the Maclaurin series for f(x) converges for all x.
At x = 0, the above Leibniz formula becomes f(n) = -2nf(n-2)
I know that f(0) = zero so...
Hey I was wondering if anyone knew of any good, intuitive and accessible versions of Newton's Principia in English? Also, if anyone could tell me of any mathematical texts written by Leibniz that are now accessible to study? Much thanks:)
Homework Statement
I'm told to find the nth derivative of a function via http://www.math.osu.edu/~nevai.1/H16x/DOCUMENTS/leibniz_product_formula_H6.pdf.
Homework Equations
Then I'm asked to show that f(n+1)=f(n)+3n(n-1)f(n-2) evaluated at x=0 when n>1
The Attempt at a Solution...
Homework Statement
The problem is attached as a picture.
Homework Equations
I believe the theories relevant to the equation are the Leibniz formula and the first or second fundamental theorem of calculus, I have two books and one lists the first theorem as the second and vice-versa...
Hi,
I'm concerned with finding the Euler-Lagrange equations for slowly modulated surface gravity waves and, as is custom in this type of physical problem, I would like to consider the averaged Lagrangian defined as
\mathcal{L}=\frac{1}{2\pi}\int_0^{2\pi}Ld\theta
where \theta is...
Hi
I have a question for change of notation.
Quote from textbook:
As an example of a singular problem on a finite interval, consider the equation
xy'' + y' + λxy = 0, (6)
or
−(xy')' = λxy, (7)
on the interval 0 < x < 1, and suppose that λ > 0. This equation arises in the study of...
I have a question I should have asked a LONG time ago.
When we see this notation being used in such formulas as i=dq/dt (definition of current) or dy/dx (etc, etc), are we saying (in the case of current) that current is equal to the change in charge with respect to time? Or is it the "current...
I know this question has been out there many times. I read many threads already but I just didn't find a satisfactory answer. What some people say is that Leibniz notation is just a notation and not a fraction. Then we treat this notation as a fraction. But what's the reason to do it if a human...
I'm trying to calculate the following function (at x=1) with accuracy of 10^(-3).
f(x)= \int^x_{0} \frac{1-cost}{t}
What I've tried:
f(1)=f(0)+f'(c)=1-\sum_{k=0}^\infty \frac{(-1)^nc^{2k}}{(2k)!}
But now I don't know how to calculate this expression. [I know that this series is convergent...
Use the Leibniz rule to derive the formula for the Lie derivative of a covector \omega valid in any coordinate basis:
(L_X \omega)_\mu = X^\nu \partial_\nu \omega_\mu + \omega_\nu \partial_\mu X^\nu
(Hint: consider (L_X \omega)(Y) for a vector fi eld Y).
Well I have the formula L_X(Y) =...
Homework Statement
If y=f((x2+9)0.5) and f'(5)=-2, find dy/dx when x=4
Homework Equations
chain rule: dy/dx=(dy/du)(du/dx)
The Attempt at a Solution
In my opinion giving f'(5)=-2 is unnecessary as:
y=f(u)=u, u=(x2+9)0.5
dy/dx= (dy/du)(du/dx)
(dy/du)= 1
(du/dx)= x/((x2+9))0.5
dy/dx =...
Hi there, I'm a new user to the forums (and Calculus) and I 'm hoping you can give me your opinion on my chain rule form below. When learning the chain rule, I was taught two forms. This form:
\frac{d}{dx}f(g(x))=f'(g(x))g'(x)
As well as the Leibniz form...
dy = lim \Deltax-->0 (f(x+\Deltax) - f(x))
dx = lim \Deltax-->0 (\Deltax)
Therefore dy/dx is f'(x) if f(x) = y
Is all of this true? I'm tired of integrating with variable substitution and not knowing what du by itself really means. People are always saying that Leibniz notation doesn't...
Hi all,
While reading through physics history book, i en counted an attempt to show the very basics of Leibniz notation; the following is shown:
If
s = (1/2)gt^2, then
s + ds = (1/2)g(t + dt)^2
s + ds = (1/2)gt^2 + gtdt + (1/2)gdt^2, then because of the first line
ds = gtdt +...
(nevermind, answered my own question after spending the time to type this up!)
Hi,
I was flipping through Hilbert's Geometry and the Imagination, and in it, he includes a proof of Leibniz' series ( pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... ) which is carried out by estimating the area of a circle at...
according to leibniz, if i have a series (An) with an alternating sign, in order for the series to converge, i need either |An| to converge, or |An| to diverge and An=0(for n->infinity) and A(n)>A(n+1)
BUT in the following series where An= ((-1)^(n-1))*1/(n+100sin(n)) the series |An|...
in the following question i am given the series:
Σ((-1)^(n-1))*1/(n+100sin(n))
and am asked if the series converges or diverges.
as far as i know Leibniz's law for series with alternating signs states that if the series of the absolute values diverges then we check the following 2...
Homework Statement
\int_0^1\frac{x-1}{\ln{x}} dx
Homework Equations
\Phi(\alpha)=\int_0^1\frac{x^{\alpha}-1}{\ln{x}} dx
The Attempt at a Solution
In the answers they say:
\Phi '(\alpha)=\int_0^1\frac{x^{\alpha}\ln{x}}{\ln{x}} dx=\frac{1}{\alpha+1}
but the derative is wrong, right? I don't...
Hello I made a youtube video trying to explain Leibniz notation because it is something I found very difficult at first and very non intuitive. There doesn't seem to be many good explanations on the internet either, leading me to believe that no one fully understands the notation.
I would like...