Technically, not simplification.

Maxwell's equations are a set of four fundamental laws that describe the electric and magnetic fields arising from varying distributions of electric charges and currents, and how those fields change in time. They are:

1. Gauss’s law (aka Gauss’s flux theorem)


2. Gauss’s law for magnetism


3. Faraday’s law of induction


4. Ampère’s circuital law with Maxwell’s correction


And together with the Lorentz force law,


Maxwell's equations form the foundation of classical electrodynamics, classical optics, and electric circuits. So, they do solve not only circuit problems, but also relativistic quantum electrodynamics problems. You'll find some interesting applications of Maxwell's equations in RADAR Cross-Section (RCS) modeling of target. Somewhere in the near future, you will love this book.



This post has been edited by Critical_Fallacy: Jan 20 2014, 05:02 PM

I don't think you get the full picture of differentiation. We DO NOT technically apply the Power Rule to an Exponent Function. This is not about math or calculus. This is about Pattern Recognition, which you must train yourself to recognize the type of a math function. The Differentiation formulas are actually shortcuts and come in handy if you can recognize the function. Otherwise, I'm afraid you have to do the hard way:

This circuit is easy but tricky.

Indirectly. Robotics is a very vast field, and open to innovations. My knowledge is limited to robot motion planning using search algorithms such as Dijkstra's algorithm, A* (pronounced "A-star"), D*, Fortune's algorithm for generating a Voronoi diagram, and Rapidly-exploring Random Tree (RRT). If you are interested in Reinforcement Learning, perhaps you'd like to read up on Approximate Dynamic Programming. I'll recommend this book, which you can find a number of interesting algorithms such as Q-Learning, TD(λ), Back-Propagation through time (BPTT).



If you are interested in Evolving Intelligent Search Engine Optimization (EISEO), Intelligent Control (IC) is a hybrid approach to Decision Support System (DSS) based on Soft Computing and Computational Intelligence methodologies. In recent years, the IC approach has been growing rapidly in visibility and importance. The guiding principle of intelligent control is that, in general, superior performance is achieved when Fuzzy Logic, Neural Networks (NN), Bayesian Networks (BN), and Genetic Algorithms (GA) are used in combination rather than in stand-alone mode. The four paradigms have their own advantages and disadvantages.

The advantage of Fuzzy Multi-Criteria Decision Making is that it works using approximate information and it represents knowledge in a linguistic form, which resembles human-like reasoning. The NN has the advantage of learning from experience. BN is a well-developed methodology for mapping real world problems into statistical formulations, and if the causal model is accurate and the evidence accurately represented, Bayesian systems give scientifically defensible results. GA optimizes or learns through adaptation of a system’s structure or parameters by evolving a random population while exploring the search space in several directions. The fusion of the four paradigms creates an intelligent system with a certain degree of autonomy and enhanced adaptive performance.

The speed in question is in fact the Mean Orbital Speed, which is equal to the distance traveled over the time taken. The question tells us the time (more accurately, Orbital Period, T) it takes for the Earth to go once around the Sun is 1 year. So in order to find the Mean Orbital Speed, we just have to figure out the distance traveled by the Earth when it goes once around the Sun. To do that, we will ASSUME that the orbit of the Earth is circular, so that we can determine the circumference.



To explain why this is Mean Orbital Speed and not its "precise" velocity, we first need to understand this is a rotational motion, which means the motion has radial () and azimuthal () components of orbital velocity ().



Because we assume the orbit of the Earth is perfectly circular, the radius (distance between Earth and Sun) is constant. This implies the radial velocity . Therefore, if this is a circular orbit, then the speed of the Earth around the Sun is



However, Kepler’s first law tells us that the planets follow elliptical paths around the sun. This is the case for Earth's orbit and the radial velocity is non-zero, which in turn implies that the orbital velocity is NOT the Mean Orbital Speed (|v| ≠ 2πr / T). But not all ellipses come in the same shape. The shapes are described by their 'eccentricity' (which tells us how flatten they are). The eccentricity (e) is a number that varies between 0 and 1, with 0 being a circle, 0 < e < 1 being an ellipse, 1 being a parabola, and e > 1 being a hyperbola. It turns out that the orbit of the Earth right now as an eccentricity of 0.016. This means it is almost a circle, making our approximation on the orbital speed 30 km/s valid.



In the two-body problem, the more precise orbital velocity of Earth can be obtained using the Vis-viva equation (beyond A-level):



This post has been edited by Critical_Fallacy: Jan 21 2014, 05:35 PM

This textbook is fairly big and thick in my library shelf.

If you want a non-rigorous mathematical treatment of Markovian Decision Process, try reading Operations Research books because the prerequisites are basic knowledge of Markov chains, Linear programming, and Probabilistic dynamic programming.

This is not a play on words, where Speed is Scalar and Velocity is Vector. In fact, the question asks to determine how fast The Earth orbits around the barycenter of a two-body system, usually around a more massive body, The Sun, in this case. In Astrodynamics terminology, this is Orbital Velocity, (see the blue-circled velocity), which the time derivative of the Earth's Position Vector, r.

This Orbital Velocity vector can be resolved into component (the green-circled velocity) along the outward radial from The Sun and azimuthal component (the red-circled velocity) perpendicular to it. At this moment, it is important to point out that . Otherwise, you're likely to confuse between the Orbital Velocity and the formula for Radial Velocity.



In Part 1, you calculate the red-circled velocity, which happens to be the Mean Orbital Speed if you assume the Orbit is perfectly circular. Because the radius r is constant, the Radial Velocity (green-circled) is zero, as explained in Post #767.

In Part 2, you should explain that in fact it is necessary to calculate the blue-circled velocity, not the red one, by arguing that Kepler’s 1st Law states that all planetary bodies (in our Solar System) follow elliptical paths around the sun.

Since the Earth's orbit is elliptical, the Orbital Velocity (blue-circled) can be determined using Pythagorean theorem as shown in Post #767, because the Radial Velocity (green-circled) is non-zero. This is due to the fact that the Sun's radial coordinate of Earth can be obtained as an angle-varying function of θ (aka True Anomaly in Astrodynamics terminology):



where h is the specific angular momentum and μ is the Sun's gravitational parameter in this case.

Not expecting you to understand everything about Astrodynamics, but the whole is greater than the sum of its parts. The failure of some Teachers to explain a knowledge correctly. The failure of some Student to learn a knowledge properly.

At your level, you should a developed analytical thinking skill. For example, what does velocity really mean for a rotating body? Are you very sure that the velocity of a rotating body is the red-circled one? If you look at the diagram carefully, you should notice that the blue-circled velocity is a tangential velocity.

Have you learned about Kepler's Laws of Planetary Motion in your Physics class?

Please show the equation of the relationship between acceleration, change of distance, and constant velocity.

As mentioned by v1n0d, first you should perform the Exactness Test on the given First-order ODE.



If it fails test, then the ODE is not exact and it cannot be solved using the normal procedure for exact 1st-order ODE. If you look at the ODE carefully, the M and N are simple enough to be expressed in an alternative ODE form:



This new form of ODE is not separable, but it can be transformed into a separable one by changing the variable. You can easily recognize that this ODE is homogeneous because all terms in the numerator and the denominator are of first degree. Because of this, the function of a homogeneous equation can be expressed as f(v) where v = y/x or y = xv.



Simplifying it:



Rearranging gives a separable ODE:



Its solution can be obtained by integration:



P.S. Pattern recognition is a very important skill in determining the method of attacking a particular type of ODE.

Has your Physics Teacher covered Projectile Motion? maximR, RED-HAIR-SHANKS, & ystiang are going to love this interactive simulation.

Click to run the simulation!


This post has been edited by Critical_Fallacy: Jan 24 2014, 03:02 PM

Phew! Master your Partial Fraction, Completing the Square, Algebraic Manipulation skills, and Integration by Substitution method...



This post has been edited by Critical_Fallacy: Jan 24 2014, 07:43 PM

Nope! Human Cannonball needs Physics simulation too (including Air Drag as well).

This Planetary Orbit Simulator shows you that the Earth's Orbital Velocity (an angle-varying vector) is NOT the same as the Mean Orbital Speed (a constant speed). Click the 4th Tab to access to the Newtonian features. Check the "Vector" box. The Blue line is a Periodic function with an extremely low Peak-to-Peak amplitude in astronomical sense. I hope this is the simple enough.

Hi Akatsuki0895, maximR, RED-HAIR-SHANKS,

This is the most Graphically Interactive Orbit Simulator developed by Laboratory for Atmospheric and Space Physics (LASP), University of Colorado. I love the Comet!



For other Interactive Lessons, click here.

This post has been edited by Critical_Fallacy: Jan 25 2014, 02:17 AM

Thanks for supporting LaTeX. You've been very helpful to delsoo and others.

Yes, you may and you'll arrive at the same answer. But, can you tell us why?

Hi Akatsuki0895, maximR, RED-HAIR-SHANKS, v1n0d, ystiang

This is my favorite simulator because it is by far the most accurate Multi-Body Orbit Simulator according to the Orbital Mechanics. The simulator shows how a planet and its star orbit a common center (the barycenter), although the Sun's wobble is barely perceptible. The most interesting "preset" would be the Sun, Planet, Comet. You must run it to see the "unexpected" for yourself.

Hi ∂soo,

This first-order ordinary differential equation (ODE) arises in the (resistor-capacitor) RC circuit during capacitor is charging.



Rearranging the equation:



There are 2 general methods to solve this ODE: (1) Heuristic, and (2) Algorithmic. By Heuristic method, you'll intuitively integrate by u-substitution, as hinted bt ystiang and quovadis123. Since I don't want to disturb your syllabus, you can use the algorithmic method by following a finite list of well-defined instructions for calculating a function, just like a computer.

A computer will convert the coefficient of the variable Q to a positive value, is this case, by factoring the denominator so that:



Now, integrating both side by using using the Integration Table (a computer will use its database):



Next, substitute integration limits,



Finally, take the inverse log of both sides and rearrange it. Thus,



The rate at which a capacitor charges depends on the product of R and C. This product is known as the time constant of the circuit and is given the symbol τ (the Greek letter tau).



For charging, the time constant, τ (tau) is defined as the time required to charge the capacitor to 63.21% of Q0. For discharging, the time constant is defined as the time required to discharge the capacitor to 36.79% of Q0. For all practical purposes, transient duration can be considered to last for only five time constants (5τ). This valuable info can be used to design a RC circuit.



For example, to design a RC circuit so that it charges in 1 sec, we set:



Using this value of 0.2, we can comfortably select R = 20 kΩ and C = 10 μF.

Tell me, how do evaluate this?