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QuantumTheory
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I thought I've seen one, but I do not understand exactly how they negate.
For example, if you have [tex]d/dx^2 + 1[/tex], does the dx cancel?
For example, if you have [tex]d/dx^2 + 1[/tex], does the dx cancel?
arildno said:What do you know about derivatives?
Nylex said:Derivatives give the rate of change of one variable with respect to another, hence they can be used to find the slope of curves. Differential equations are equations involving derivatives and have lots of applications in physics. Newton's 2nd law can be expressed as a second order differential equation: F = m (d^2x/dt^2). In electric circuits, we can use differential equations to find out how current changes as a function of time, for example in a circuit involving an inductor. There's also the Schrodinger equation, in quantum mechanics that can tell us about energy for a quantum system. The diffusion and wave equations are other examples.
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.
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 the operation of differentiation.
Cheers.
Integral said:More generally, Differential equations are basis for every meaningful physical theory in existence. They are the starting point for the math. DEqs are the mathematical language to express how things change. All of our physical observations are in the terms of how things change. If you translate the observed changes in the physical world carefully into mathematics you have a differential equation. Solve the differential equations and you have a tool to make a physical prediction.
A differential equation is a mathematical equation that relates a function with its derivatives. It describes the relationship between a function and its rate of change, and is often used to model various physical, chemical, and biological systems.
An ordinary differential equation involves a single independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are used to model systems that change over time, while partial differential equations are used to model systems that change over multiple dimensions.
Differential equations are widely used in physics, engineering, and other sciences to model and predict the behavior of systems. They are used to study the motion of objects, heat transfer, population growth, and many other phenomena.
Some differential equations can be solved analytically, meaning that an exact solution can be found using mathematical techniques. However, many differential equations are non-linear and cannot be solved analytically, requiring numerical methods to approximate solutions.
Differential equations are an important tool in mathematics for understanding the behavior of systems and making predictions. They also have applications in other areas of mathematics, such as geometry and topology. Many fundamental laws and principles in physics, such as Newton's laws of motion, can be described using differential equations.