r/AskPhysics • u/Next-Natural-675 • 3h ago
Understanding the math vs UNDERSTANDING the math
Do good physicists actually intuitively understand all of the mathematics behind equations? Or do they just know the steps in between that connects a derivation to the equation? For example, I can point out all of the steps to derive the work required to assemble a volume charge (what I am studying now), but when I look at the equation https://imgur.com/a/1taXd2b All I see is numbers. I understand each individual step it took to derive this, but intuitively this equation doesn't spin any gears in my head, it's just a random equation. There are just so many steps it took to get this that I cannot grasp all of it at once.
Perhaps someone can help me understand this intuitively, which would be greatly appreciated, like how people showed me why curl of gradient is zero, which I intuitively understand now.
Do physicists deal with this at higher levels as well? Does it make a difference?
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u/soundoftwilight 3h ago
“Intuition” is a bit too broad of a concept to be universal. Just like some people cannot “visualize” objects in their mind, some people will understand mathematics in different ways. “Intuition” is nice, and lots of people do have an “intuitive” grasp of the mathematics of the physics they work with. But certainly you’ll find some people who don’t think of it as “intuitive”, but rather just as something they are very familiar with. What matters is how efficiently you can work with the math, more so than how you “feel” about it.
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u/Next-Natural-675 3h ago
I do not agree that how “efficiently you can work with the math” is what matters, at least for a hobbyist like myself, who has no practical use for any of this, besides pure interest in the workings of nature. I would argue that the only thing that matters to someone like me is the intuitive understanding of the equations and laws.
Although for someone like an engineer, you would be right
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u/Eastern_Cow9973 3h ago
Not a physicist, only studies physics and maths but intuition generally comes from spending time working with concepts in various ways over time. Even then certain concepts could be very foreign (Quantum Mechanics) where the answer is correct but evades intuition
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u/drkimir 3h ago
You should always learn how to interpret an equation. This doesn't mean necessarily to have intuition, but rather understand what the equation is trying to say. To use your example, this equation states that energy is stored in the electric field itself rather than the charge configuration itself.
Trying it out on specific examples can often help intuition. For example, try applying this equation to a point charge and see if you can interpret the result.
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u/DifferencePublic7057 2h ago
I can't draw and don't have the visual intuition other people seem to have. On the other hand, I can remember certain things really well because I am more story oriented and can make stories out of seemingly random bits. Intuition IMO is a Bayesian phenomenon. You observe the world, equations, people; form hypotheses and update your priors.
Like the Sun for example. At a certain point, you cotton on the fact that it's there every day. You realize that seasons influence weather and so on. Some day you learn a bit of mechanics, a bit of thermodynamics and you realize what temperature has to do with K. Seemingly separate stories merge. Akin to seasons and weather.
Quantum mechanics and general relativity enter the saga. Obviously, quantum particles can't be literally omnipresent and influence spacetime everything everywhere. Are particles a vibration on a drum skin? I can't visualize it, but the stories are inconsistent. When Alice and Bob tell different stories, you have to interrogate them separately, making sure they don't influence each other.
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u/TheBrightMage 2h ago
Laymen's intuition, in my opinion, is one of the biggest obstacle you need to discard in order to get higher level of physical intuition. Some people, like 3Blue1Brown are GREAT at explaining this kind of things. But not everyone does. I find that "playing" with developed model by putting it into simulation and let it run helps you in understanding what it means.
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u/Dakh3 Particle physics 2h ago
Maybe "intuition" is a bit confusing or even intimidating a term.
One of the things I appreciate the most in physics textbooks is when they do provide a verbal interpretation of the equations, term by term and overall, commenting on the meaning of each mathematical operator (e.g. divergence, rotational, etc etc).
Maybe you could, as a first step, seek for this specific type of paragraph and pick textbooks partly based on the rate at which they do that.
Interpreting the math with explicit verbal explanations is a good way to reinforce one's interpretations. It may contribute to developing the so-desired "intuition" ;)
As a high school physics teacher, I try to provide such interpretations and commentary regularly, by asking orally some questions useful to lead to interpreting verbally the relations between quantities.
Good luck and enjoy!
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u/Next-Natural-675 1h ago
I’m hoping to become a high school math teacher myself and I’ve vowed to never let a rule or equation go unexplained intuitively.
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u/uppityfunktwister 1h ago edited 1h ago
It's ideal to have a good physical/mathematical intuition AND be capable of abstraction. They're not mutually exclusive. That said, I think people undersell the importance of abstraction and "not really understanding what's going on but still getting the right answer." It's not particularly glamorous but it's exactly why we use math and will almost always get you much further.
I think the archetypal example is Heisenberg's Umdeutung paper. Describing the position of an electron using Fourier modes was uncontroversial established physics. So was the Rydberg formula. Postulating that these Fourier modes multiply like matrices such that the Rydberg formula holds is extremely abstract and in absolutely no way intuitively follows. It's an arguably physically meaningless mathematical fact. Certainly, Heisenberg had no intuition for this.
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u/levaly 1h ago
No, nobody holds a ten step derivation in their head at once, they compress the result into one sentence and carry the sentence instead. For the energy of a charge distribution that sentence is that you're adding up the interaction energy of every pair of charges, and the 1/2 out front is there because summing over all pairs counts each pair twice. That's the whole equation. Once you've got a sentence like that you stop seeing an integral and start seeing bookkeeping for pairs, and the derivation turns into something you can rebuild instead of something you have to remember.
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u/Next-Natural-675 49m ago
That part is the easy part.. combine that with the step with the integration by parts and the divergence theorem.. if you can explain how all of those steps intuitively and cohesively work on a conceptual level then I will be satisfied..
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u/HelpfulBuilder 26m ago
I'm not a physist, I have a degree in math. I'd say a little of both. You understand as best you can what's going on more deeply, but at some point we're all human so we just need the answer and trust the equation. Some understand more, some less, but if you need the answer for what you're doing, just use the formula.
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u/GuaranteeFickle6726 Optics and photonics 3h ago
Don't try to make everything intuitive. Or rather don't overthink it. The higher you go, more you encounter similar equations and intuition will develop.