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For example, if you have [tex]d/dx^2 + 1[/tex], does the dx cancel?

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For example, if you have [tex]d/dx^2 + 1[/tex], does the dx cancel?

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arildno

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What do you know about derivatives?

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arildno said:What do you know about derivatives?

Quite a bit, I understand that deriviatives are an infinitely small section of a curve, making up the entire curve with an infinite amount of infinltey small sections.

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Integral

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Nylex said:

I'm sorry,i'd have to contradict you here.The mathematical expression for Newton's second law (linear motion) is actually:

[tex] \frac{d\vec{p}}{dt} = \sum_{i} \vec{F}_{i} [/tex]

,where 'i' is an index which takes values in a subset of N.

Try to figure out why i've written it in such form.

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dextercioby said:I'm sorry,i'd have to contradict you here.The mathematical expression for Newton's second law (linear motion) is actually:

[tex] \frac{d\vec{p}}{dt} = \sum_{i} \vec{F}_{i} [/tex]

,where 'i' is an index which takes values in a subset of N.

Try to figure out why i've written it in such form.

but how can you express the force on an object. It can be expressed as

d^2 (x) / d (t^2).

- harsh

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Tide

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That's true only when the mass is constant in which case

[tex]\frac {d\vec p}{dt} = \frac {d m \vec v}{dt} = m \frac {d \vec v}{dt} = m \frac {d^2\vec x}{dt^2}[/tex]

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[tex]

\vec{F}_{net} = \frac{d}{dt}(\vec{p}) = \frac{d}{dt}(m\vec{v})

[/tex] whereby [tex]\vec{F}_{net} = \sum_i{\vec{F}_i}.[/tex]

For the non-relativistic case ([tex]v << c[/tex]), we can add the approximation that the mass is a constant and treat it as such. Of course, for relativistic speeds (usually [tex]v \ge 0.1c[/tex]), we need to invoke the special relativity mass correction factor of [tex]m = m_0\frac{1}{\sqrt{1-(v/c)^2}}.[/tex]

How this all applies to differential equations is that the physical phenomenon of nature are modeled by differential equations. Newton's second law above gives a great example. Take the motion of a mass resting on a flat frictionless tabletop that is displaced from its equilibrium position by some deflection [tex]x[/tex]. Now, the primary force we're concerned with is the restoring force of the spring, given by [tex]F = -kx[/tex] where I drop the vector symbols because we only have 1-dimensional freedom of movement and the direction is indicated by sign. Now, by Newton's second law, this gives us the following equation:

[tex]

F_{net} = \frac{d}{dt}(mv) \implies -kx = mx'' \implies x'' + \frac{k}{m}x = 0.

[/tex]

Voila. This is a linear second order differential equation with constant coefficients that models the motion of the mass. Two solutions of this differential equation are:

[tex]x(t) = \sin{\left(t\sqrt{k/m}\right)}[/tex] and [tex]x(t) = \cos{\left(t\sqrt{k/m}\right)}[/tex]

and by an elementary theory from differential equations, we know that all solutions of the system can be written in the form of

[tex]x(t) = C_1\sin{\left(t\sqrt{k/m}\right)} + C_2\cos{\left(t\sqrt{k/m}\right)}[/tex]

where [tex]C_1[/tex] and [tex]C_2[/tex] are constants determined by the initial conditions of the system. A knowledge of physics and differential equations combined is a powerful tool for producing accurate models of the physical universe.

As for your comment about derivatives and infinities, this is bordering into the philosophy of mathematics, which has a very long history. Debates around infinity and infinitesimals still go on today despite our everyday application of them. Strictly speaking, mathematicians use the term "differential" to describe small changes in one quantity (e.g. [tex]dx[/tex]) whereas they use the term "derivative" to refer to the

Cheers.

---

Mike Fairchild

http://www.mikef.org/

"Euclid alone has looked on beauty bare."

--Edna St. Vincent Mallay

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Tide

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It's possible for the mass to be a variable even for strictly classical physics problems such as in the case of a rocket expelling mass as it accelerates.

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mjfairch said:Tide, harsh, and dextercioby are all basically saying the same thing, but let me provide some elucidation. As a historical note, Newton wrote his second law originally in terms of change-of-momentum. That is, the net force acting on an object causes a change in momentum as follows (where the net force and momentum are both vector quantities):

[tex]

\vec{F}_{net} = \frac{d}{dt}(\vec{p}) = \frac{d}{dt}(m\vec{v})

[/tex] whereby [tex]\vec{F}_{net} = \sum_i{\vec{F}_i}.[/tex]

For the non-relativistic case ([tex]v << c[/tex]), we can add the approximation that the mass is a constant and treat it as such. Of course, for relativistic speeds (usually [tex]v \ge 0.1c[/tex]), we need to invoke the special relativity mass correction factor of [tex]m = m_0\frac{1}{\sqrt{1-(v/c)^2}}.[/tex]

How this all applies to differential equations is that the physical phenomenon of nature are modeled by differential equations. Newton's second law above gives a great example. Take the motion of a mass resting on a flat frictionless tabletop that is displaced from its equilibrium position by some deflection [tex]x[/tex]. Now, the primary force we're concerned with is the restoring force of the spring, given by [tex]F = -kx[/tex] where I drop the vector symbols because we only have 1-dimensional freedom of movement and the direction is indicated by sign. Now, by Newton's second law, this gives us the following equation:

[tex]

F_{net} = \frac{d}{dt}(mv) \implies -kx = mx'' \implies x'' + \frac{k}{m}x = 0.

[/tex]

Voila. This is a linear second order differential equation with constant coefficients that models the motion of the mass. Two solutions of this differential equation are:

[tex]x(t) = \sin{\left(t\sqrt{k/m}\right)}[/tex] and [tex]x(t) = \cos{\left(t\sqrt{k/m}\right)}[/tex]

and by an elementary theory from differential equations, we know that all solutions of the system can be written in the form of

[tex]x(t) = C_1\sin{\left(t\sqrt{k/m}\right)} + C_2\cos{\left(t\sqrt{k/m}\right)}[/tex]

where [tex]C_1[/tex] and [tex]C_2[/tex] are constants determined by the initial conditions of the system. A knowledge of physics and differential equations combined is a powerful tool for producing accurate models of the physical universe.

As for your comment about derivatives and infinities, this is bordering into the philosophy of mathematics, which has a very long history. Debates around infinity and infinitesimals still go on today despite our everyday application of them. Strictly speaking, mathematicians use the term "differential" to describe small changes in one quantity (e.g. [tex]dx[/tex]) whereas they use the term "derivative" to refer to theoperationof differentiation.

Cheers.

I'm sorry but i wasn't referring at all to the relativistic case (and maybe neither Arildno),but to a very classical case,analyzed at the end of the 19-th century (round 1892,if i'm not mistaking) of the Newton's equations.

Does the name Tsiolkowskiy tell u something????????????Guess not,else u would have realised i was referring to his famous equation which describes the movement of a rocket.Yes,you saw well,the first rockets ever launched were the famous V1 and V2-s of the German Army (Wehrmacht) in 1941 in the Battle of England (actually they launched a much more massive attack later,towards the end of thewar).Anyway,the theory of rocket trajectories had been done many years before (round 50) by the russian Tsiolkowskiy.He bassically assumed that Newton equation would be written for a classical body with varying mass,viz.

[tex] \frac{dm(t)}{dt}\vec{v}+m(t)\frac{d\vec{v}}{{dt} =m{t}\vec{g}+\vec{F}_{friction forces} [/tex]

Daniel.

PS.I guess he assumed the friction forces Stokes type.

EDIT:Arildno was referring at the same thing. :tongue2:

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Cheers.

---

Mike Fairchild

http://www.mikef.org/

"Euclid alone has looked on beauty bare."

--Edna St. Vincent Mallay

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saltydog

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Integral said:

Go get um' !

SD

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mathwonk

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I have a small question. Is 'Differential Equation' part of 'Calculus'?

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As a result,I think this is a reson of Differential Equations.

Sorry for bad English.

+

I want to ask you a question..

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html here is the solution of capacitors charge from differential equations.I want to ask you,is there any general solution for capacitors charge or another things? In other words,does it mean that some def equations is the single solution of physical situation?

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