Pictures of abstractions behind mathematical equations?

[jimgavagan]

I need to understand physics (and other) concepts but I'm having trouble visualizing what the equations mean.

One example would be an applied math thing - I want to understand what about electromagnetism exactly the Maxwell equations are describing exactly so I can gain insight into what math equations are describing exactly when they refer to the physical world - I then also want to understand how the combining the equations for magnetism and the equations for electricity lead to the math-based discovery of electromagnetism (Maxwell equations) in order to gain insight into what math-based discoveries of physical phenomenon look like, just for its own sake.

(Ultimately, though, once I figure out through the examples above what about the physical world that mathematical equations are describing, I want to use this info in order to make my way into the even more abstract world of theoretical math, if that helps with your replies.)

So I can start visualizing math better...Are there any websites or anything with pictures of mathematical concepts/abstractions given a particular equation, or even, perhaps, given just a particular branch of math in general?

Thanks!

 

[tiny-tim]

hi jimgavagan!

… I want to understand what about electromagnetism exactly the Maxwell equations are describing exactly so I can gain insight into what math equations are describing exactly when they refer to the physical world …

the first pair of https://www.physicsforums.com/library.php?do=view_item&itemid=91" (Gauss' law and Gauss' law for magnetism) essentially say that electricity (or magnetism) are like a conserved fluid …

magnetism is like a fluid with no sources and sinks: the amount of fluid is conserved, ie the amount flowing into any region always equals the amount flowing out: divB = 0

electricity is like a fluid which does have sources and sinks (like taps and plugholes): positive charges are sources, and negative charges are sinks: a source or sink produces a constant inflow or outflow of electric field, equal to the charge: the amount of fluid is conserved, once we take into account the sources or sinks: ie the amount flowing into any region always equals the amount flowing out plus-or-minus the rate from the sources or sinks (the charge inside the region): divE = ρ​

the second pair of Maxwell's equations (Faraday's law and Ampere's law) are essentially energy laws: they say that electricity and magnetism are not a https://www.physicsforums.com/library.php?do=view_item&itemid=174": potential energy is supplied to the system, proportionally to the rate of flow of field and sources (or sinks) …

since https://www.physicsforums.com/library.php?do=view_item&itemid=269" is measured by its effect along a path, we always integrate round a closed path (and differentiating that gives us a curl) …

they say that potential energy is continually supplied to the system, proportional to the rate of flow of the electric and magnetic fields, and of the electric charge (rate of flow of electric charge, ∂q/∂t, is of course the same as electric current, I; and there is no magnetic charge)

however, it is mixed-up :

the electric potential energy is supplied from (minus) the flow of the magnetic field: ∫E.dl = - ∂/∂t ∫∫B.dA,
and the magnetic potential energy is supplied from the flow of the electric field and of the electric charge: ∫B.dl = ∂/∂t ∫∫E.dA + ∂q/∂t​

 

[lavinia]

I need to understand physics (and other) concepts but I'm having trouble visualizing what the equations mean.

One example would be an applied math thing - I want to understand what about electromagnetism exactly the Maxwell equations are describing exactly so I can gain insight into what math equations are describing exactly when they refer to the physical world - I then also want to understand how the combining the equations for magnetism and the equations for electricity lead to the math-based discovery of electromagnetism (Maxwell equations) in order to gain insight into what math-based discoveries of physical phenomenon look like, just for its own sake.

(Ultimately, though, once I figure out through the examples above what about the physical world that mathematical equations are describing, I want to use this info in order to make my way into the even more abstract world of theoretical math, if that helps with your replies.)

So I can start visualizing math better...Are there any websites or anything with pictures of mathematical concepts/abstractions given a particular equation, or even, perhaps, given just a particular branch of math in general?

Thanks!

I know of no better book than Feynmann's Lectures on Physics Volume 2

 

[jimgavagan]

Hello guys,

I was looking for more like an actual picture? Thanks for these replies though, they probably can't hurt!

Jim