What is Laplace: Definition and 1000 Discussions

Pierre-Simon, marquis de Laplace (; French: [pjɛʁ simɔ̃ laplas]; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.
Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries.
He was Napoleon's examiner when Napoleon attended the École Militaire in Paris in 1784.
Laplace became a count of the Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.

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  1. P

    What Is the Steady State Value of U with Both R and V as Unit Steps?

    Homework Statement Consider this control system below: R = set point E = remaining error V = interference My question is, if both R and V are unit step \frac{a}{s}, what will the value of U be when time t\rightarrow\infty ? Homework Equations This question is based on the...
  2. I

    Convolution theorem and laplace transforms

    Okay, so this is the first time I'm encountering this theorem and I'm not very strong in calculus. But I tried to understand it myself but couldn't. Convolution theorem is the one in the attachment as give in the book ( couldn't find a way to type that out easily). My doubt is if laplace(f)...
  3. R

    Inverse Laplace Transform of (1/(s+s^3))?

    Homework Statement What is the inverse laplace transform of (1/(s+s^3))? Homework Equations The Attempt at a Solution I looked it up on wolframalpha and got 1-cos(t), but I don't understand how they got that answer. I looked up a basic laplace transform chart and didn't see anything...
  4. G

    Laplace Equation, Non-Concentric Spheres

    1. Homework Statement General solution for eccentric spheres, smaller sphere (radius, b) completely embedded within larger sphere of radius c. The centers of both spheres lie on z-axis, distance a, apart (note: c>b+a). Problem is symmetric, so consider θ=[0,∏], r=[0,c]. The inner sphere is...
  5. G

    Laplace problem for eccentric spheres

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  6. S

    Laplace equation in a square with mixed boundary conditions

    The length of the side of the square is a. The boundary conditions are the following: (1) the left edge is kept at temperature T=C2 (2) the bottom edge is kept at temperature T=C1 (3) the top and right edges are perfectly insulated, that is \dfrac{\partial T}{\partial x}=0,\dfrac{\partial...
  7. M

    How to find inverse Laplace transform?

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  8. M

    Laplace Transform of the product of two functions

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  9. G

    Is my unit step function correct for solving this differential equation?

    Homework Statement y''-4y'-32y={1 when 0<=t<1 and 0 when 1<=t y(0)=y'(0)=0 Homework Equations The Attempt at a Solution s2L(y) -4sL(y)-32 L(y)=u1(t) I am just struggling to figure out if my unit step function is correct. Solving for L(y) I get: (e-s) / (s(s2 -4s-32))...
  10. G

    Solving Laplace Transform: Need Help

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  11. T

    Finding the inverse laplace transform of (2/(s+2)^4) using Convolution theorem.

    Homework Statement Find the inverse laplace transform of (2/(s+2)^4) using the given table of identities: Homework Equations Here are the given identities: The Attempt at a Solution Alright, I realize that there is a simple identity that I can use with a factorial symbol, but this...
  12. D

    Series form of the Laplace transform

    I thought it would be obvious, but I can't find a series representation of the Laplace transform. I'm looking for something analogous to the Fourier series and how it can be used to derive the Fourier transform. I though it would simply be f(x) = \sum_{s=-\infty}^{\infty}{C_{s} e^{sx}} , but...
  13. E

    ROC and its relation to the inverse Laplace transform

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  14. L

    Intuition about definition of laplace transform

    why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how...
  15. N

    Laplace transform of the dirac delta function

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  16. L

    Laplace transform of a function squared, help with this system

    Homework Statement Use Laplace transform to the system: \frac{dy}{dt} + 6y = \frac{dx}{dt}3x - \frac{dx}{dt} = 2\frac{dy}{dt} x(0) = 2 ; y(0) = 3 The Attempt at a Solution I've tried everything on this one. I first solved \frac{dy}{dt} + 6y = 2\frac{dy}{dt} and I got y = 3e^{6t} ...
  17. fluidistic

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  18. fluidistic

    Laplace transform to solve an ODE

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  19. E

    Solving a Second Order Differential Equation with Laplace Transform

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  20. L

    Laplace transforms of a reciprocal derivative.

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  21. J

    What is the Steady State Solution for the Heat Equation in an Annulus?

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  22. H

    Unilateral and Bilateral Laplace Transform in Solving Differential Equations

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  23. M

    Are Laplace Transform Limits Equivalent to a Limit at Infinity?

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  24. D

    Laplace Transform of Delta Function

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  25. N

    Using Laplace Transform to solve a differential equation

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  26. M

    Solving Laplace Transform of cos2at | Simple Homework Guide

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  27. baby_1

    Convert a Laplace Function to image & real part

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  28. F

    Laplace transform, sum of dirac delta

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  29. 6

    Laplace Equation Solved by Method of Separation of Variables

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  30. baby_1

    Engineering What Does A=0 or S=-3/2 Indicate in Laplace Transform Analysis?

    Laplace Error in a Circuit ! Hello as you see this circuit i want to find the v(t) across the 2ohm resistor.(VC(0-)=0) i assume A node above the capacitor , so we have (in Laplace) so i have so A=0 ! or S=-3/2 ! what can we understand about A=0 or S=-3/2 ? what is their inverse...
  31. chisigma

    MHB A Laplace Transform: Integral Calculation

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  32. fluidistic

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  33. N

    Inverse Laplace Transformation of arctan (s/2)

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  34. fluidistic

    Laplace transform of a piecewise function

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  35. B

    Laplace equation w/ dirichlet boundary conditions - Partial Diff Eq.

    Homework Statement The steady state temperature distribution T(x,y) in a flat metal sheet obeys the partial differential equation: \displaystyle \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2} = 0 Seperate the variables in this equation just like in the...
  36. M

    Laplace transform converge

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  37. E

    Inverse Laplace transforms with quadratic factors

    Homework Statement This is a practice problem for a test on Laplace transforms Find L^-1[ (9s^3+17s^2+66s+45) / (s^2+9)(s+2)^2 ] (L^-1 = inverse laplace transform) Homework Equations From Laplace transform tables: L^-1[ 1 / s-α ] = e^αt L^-1[ s / s^2+ω^2 ] = cos(ωt)...
  38. B

    Laplace Sine Transform: Re[s]>0

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  39. E

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  40. D

    MHB Solution of Laplace Spherical Equation with Boundary Conditions

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  41. H

    Using Laplace Equation in one dimension to solve for a charge-free slab

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  42. E

    How to Solve Inverse Laplace Transform: Factor or Use Complex Methods?

    I have to take the inverse laplace transform of the above function. Now, I know that I can factor (s^2+5s+6) as (s+3)(s+2) and take the easy way out. However, I did it as above on a test, getting A = -1, B = -1, and C = 1. I then took the inverse laplace transform and got something involving...
  43. M

    Proving the Convolution Theorem for Laplace Transform

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  44. C

    Having Trouble with Solving a Laplace Integral? Any Pointers?

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  45. K

    Simple Laplace Transform Question (Differential Equations Class)

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  46. W

    Inverse Laplace Transforms without Prefix

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  47. O

    When to use Gen. Fourier, F.Sine, F.Cos, and Laplace Transforms

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  48. P

    Laplace Transform Homework: Solving f(t) Piecewise

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  49. A

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  50. D

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