SciRev. X102 Exam 3 wk. 9 Maths ( 24 questions with verified solutions )

The Enlightenment

-a movement that emphasized science and reason as guides to help see the world more clearly

-emphasized freedom of thought and action w/out reference to religious/traditional authority

-advocated use of reason to understand nature, law, economy, etc

-rejected revealed religions (as christianity) but accepted the existence of a God (=deism)

Mathematical Principles of Natural Philosophy

-newton

-relied extensively on geometrical diagrams

In the 18th century, mathematics became more...

-algebraic, thus relying on complex differential equations, profressively using fewer diagrms

-Lagrange proudly state in his 1788 Analytical Mechanics, that his treatise did not contain a single disgram

Principia mathematica

-newton

-showed that the force towards the the focus S of an elliptical orbit was inversely proportional to the square of the distance SP

-shows Newton's proof of Kepler's equal-area law of planetary motion; the proof proceeded entirely by geometry of "synthesis"

mechanique analitique

-Lagrange

-contain's Lagrange's equations

-the victory of analysis over synthesis is complete

-these mechanics take the place of Newton's three laws

Gottfried Wilhelm Leibniz

-a polymath

-was the first to publish on the differential and integral calculus

-like newton, he had a huge influence in the 18th century

leibniz on calculus and computing

-published the first papers on differential and integral calculus in one of the first scientific journals of the time (The Acts of Erudite Men)

-Leibniz also designed one of the first mechanical calculating machines

Mathematical curves were thought of as what in the 17th century

defined and conceived in terms of their geometric properties or construction methods

Mathematical curves were thought of as what in the 18th century..

-shifted into thinking in terms of differential equations

-problems posed could only be solved by the manipulation of such equations (such as integration)

-curves that were investigated included the catenary

-Bernoulli

catenary

-the curve traced by a hanging chain

-picture is a type of inverted catenary

Bernoulli

-learned and developed the new mathematical techniques in the 18th century

-first to use Lebniz's calculus in mechanics (was followed by his brother Johann and his nephew Daniel)

One of the first prominent female mathematicians?

-The Marquise Emilie du Chatelet

-produced a french translation of Isaac Newton's Principia w/ significant additions and calculations

-published "Institutions of physics" which spread Leibnizian ideas

-worked w/ Voltaire

-was faithful to newtonian ideology

-looked favorably on vis viva (much to Voltaire's dismay)

New Mechanical Principles

-rigid body

-fluid bodies

-elastic/flexible bodies

rigid body

-a body w/ a mass distributed in space

-its motion is the combination of the motion of its center of mass and a rotation of the rigid body around the center of mass

-objects that collide must instantaneously change their velocities

-can't pass thru intermediate velocities

-polygon curve

-blow

fluid bodies

-compressible (air) or incmopressible (water)

elastic and flexible bodies

-elastic = springs

-flexible = strings

-type of motion requires a new type of equations

-would be deformed

-would reound when they collide, therefore changed in their velocity would be continuous

-rigorous curve

Leonhard Euler

-worked in all areas of pure and applied mathematics, including mechanics

-euler formulated the mechanical principles for dealing w/ rigid bodies

-worked extensively in celestial mechanics as well

Jean-Baptiste le Rong d'Alembert

-was involved in the vis viva controversy; a dispute about which quantity is conserved in the universe

vis viva

- = living force = mv2

-was eventually chosen

-this notion was generalized as the prinicple of conservation of energy in the 19th century

vis viva vs action

-vis viva = living force

~guaranteed that the universe would never run down

~is the dynamic quality that was conserved in the universe

~measure of God's desire to converse his creation

- action = measure of God's efficiency

~the 'action' in any motion should be a minimum given God always acts in the most efficient, econimic ways

~became the foundation of analytical mechanics

D'Alembert and Denis Diderot

-edited a defining work of the Enlightenment, the "Encyclopedie"

most decisive progress of the vis viva vs action came from an experiment. What was the experiment?

-dropping balls of different masses but the same size onto clay to evaluate the impressions that were made

-To ger similar impressions they found that is was necessary to drop the balls from heights inversely proportional to their masses

Who set rational mechanics on its course?

newton (Lagrange carried it on; boasts in his book)

who was in support of rigid/hard bodies?

-newton and atomists, D'Alembert, Maupertuis...

-Lebniz, Bernoulli and Euler denied the existence of hard bodies (insisting all change must be continuous)