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): postive 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

 

[blue_raver22]

I completely understand your struggle to visualize mathematical concepts and equations. It can be challenging to grasp abstract concepts without a visual representation. However, I want to assure you that with practice and a deeper understanding of the underlying principles, you can develop a strong intuition for these equations.

One way to approach this is by breaking down the equations into smaller, more manageable parts and understanding each component individually. You can also try to connect the equations to real-life scenarios or physical phenomena. For example, in the case of electromagnetism, you can think about how the equations relate to the behavior of a compass or the working of an electric motor. This will help you gain insight into the physical meaning behind the equations.

Additionally, there are many resources available online that provide visual representations of mathematical concepts. You can find websites, videos, and even interactive simulations that can help you visualize and understand the equations better. Some popular websites include Khan Academy, Wolfram MathWorld, and MathisFun. These websites offer a wide range of topics and provide step-by-step explanations with visual aids.

It is also helpful to seek guidance from a math or physics tutor who can help you understand the equations and their applications in a more personalized and interactive manner. They can also provide you with additional resources and practice problems to deepen your understanding.

In conclusion, while it may seem daunting at first, with patience, practice, and the right resources, you can develop a better understanding of mathematical concepts and equations. Keep exploring and asking questions, and you will eventually gain a strong intuition for these abstract concepts.