For example, Maxwell’s equations contain upside down triangles. What exactly do they represent?
How does it apply to the real world?
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It represents a vector derivative.
It tells you how a vector (a quantity that has both an amount and a direction) will change over time.
For instance, in some of Maxwell’s equations, it represents how a charged particle’s movement is affected by electromagnetic fields over time.
The upside triangle (called ‘del’) indicates the gradient.
When you follow del with a dot, it means divergence. When you follow del with an ‘x’, it means curl.
These are all concepts in multi-dimensional calculus. The gradient is simply a list (also called a ‘vector’) of all the partial derivatives of a functions – essentially the slope of the function in the direction of one of the variables. So if you have a function in x, y, z, the gradient would be a list containing the slope of the function in the x direction, the slope of the function in the y direction and the slope of the function in the z direction.
For divergence, think of water gushing out of a faucet. If you imagine a plane through which that water is passing, you want to know how fast/much is passing through that plane. For this, you need to calculate the ‘flux’ of the water stream through the plane and you use the divergence for that.
Curl is similar except it deals in rotation. You’re looking to find out whether the water is rotating – if you were to toss a cork into the water, would it just go one direction or would it spin around?. This is what the curl tells us.
However, because we’re dealing with complicated phenomenon, we’re not representing that water flow as a single number but rather a function which tells us what every bit of water is doing at any given time. Thus, the curl and divergence are themselves functions that tell us what is happening at every point in space.
They derivatives or rates of change.
Vectors have more moving parts than numbers, so there’s a bit more variety in the types of derivatives that are useful.
In Maxwell’s Equations, they tell you how electric and magnetic fields either push in/out or swirl. If the upside down triangle has a dot, that’s about how it pushes in/out, if it has multiplication symbol then that’s how it swirls. This constitutes complete information about how electromagnetic fields work.
Maxwell’s Equations are compact, abstract formulae that tells us how electromagnetism works at a fundamental scale (classically, at least). It’s like an intricate piece of origami that can be unfolded in innumerable ways. If you want to use them in real life, then you need to know how to unravel them for the specific concrete situation that you’re in. This is where multivariable calculus, partial differential equations, mechanics, statistics, and computer modelling help. With these, you can take the compactified instructions of the Maxwell equations and slowly unwrap them to conform to the constraints that you’re working in in order to deduce concrete conclusions.
I’m going to try and answer your second question:
People like you and like me, who are not very fluent in advanced mathematics rely on natural language to describe the “real world”. This inherently circumscribes what we can describe because natural language has evolved to allow us to describe what we are capable of perceiving through our senses. On the other hand, mathematics has evolved to describe the “real world” that exists beyond our senses. The most obvious (to most of us anyway) part of the “real world” that natural language is unable to fully describe is related to various forms of energy and mass. We have words to name an electron but words will never full describe what an electron is or does. The same is true of all sub-atomic properties and entities as well as forces such as magnetism and gravity that are macroscopic but invisible. The “real world” as described by mathematics is largely unavailable to us without the use of mathematics.
Del dot is divergence, which you can visualize as field lines diverging outward from (or converging inward to) a point. Electrical field lines diverge from charge density, which means that field lines go to and from electrical charges. The zero in the corresponding magnetism equation means there are no magnetic charges or “monopoles”.
The curl (del cross) of electric fields is determined by the presence of changing (d/dt) magnetic fields, and vice versa. So when you have a changing electric field, it induces a circular magnetic field around it, and vice versa. Magnetic fields are also induced by currents J, or moving charges, which is shown in the real world by magnetic fields induced around electrical wires. You see this in electromagnets.
Imagine that you have a smooth wooden floor and you roll a ball across it. The ball moves in certain predictable ways. It is easy to picture where is the ball will go when you bump it. It is easy to calculate the equations of motion in the situation.
Now take the same ball and play with it on a curved hillside. The ball does not act the same way it did on the flat floor. If you push it uphill it turns around and goes back downhill, or curves a certain way, or tries to go downhill very quickly. If it is a big hill, and you leave the ball rolling, it might be going super fast by the end of it. There might even be differences in terrain, such as gravel or smooth grass. The equations to control how the ball is moving suddenly got much, much more complicated because gravity is pulling it, but how the hill is shaped changes how gravity moves the ball as well as your touches.
What would happen if you empty a big bucket of balls out on a certain spot of the hill? Most of them would roll down the hill the same way, but this would look different if you dump the balls out on different spots of the hill.
The upside down triangle (del) It is a way of handling similar forces with electricity and magnetism. It isn’t super concerned with one ball at a time, but it helps describe how lots of balls act on a certain spot on the hill. In the case of Maxwells equations, it is more concerned with how electromagnetic fields change and push and pull differently at different spots.
Which way is “upside down” for a triangle?
Its the nabla operator. An operator is just a thing that codes an operating.
For example if all you want is to multiply a number a by a number b then you can describe an operator that by definition does this a × b. You can have another number c and you use the operator on it which gives you a × c. Lets call this operator M_a for multiplication. So (M_a)b = a × b. Simple enough.
If your operating is a bit more complicated like take the partial derivitives of a scalar field its a bit painful to write down all the derivitives. We often use a gradient (nabla operator) when we have a force filled. You can often write your force filled up with just a scalar field thats only a number assigned to all points in usually 3D sapce. But a force field assigns vectors to all points in space. So how do we turn numbers into vectors. (It only works with conservative fields. Like gravity or the electricfield.)
So lets say we use an xyz coordinate system. Now our scalar or potential filed P is a function of xyz so P(x,y,z). What we do is this: (dP/dx, dP/dy, dP/dz) we build a vector out of the partial derivitive of P which is exactly the vector of the force at that point. This operation of take the partial derivitives of the function and plug them in a vector is often compressed into symbols like that upside-down triangle.
For a gradient like this grad(P) is often used. There are other useful operations you can do to a filed like a vector filed. Like divergence often used as div(v) or nabla • v or rotation used as rot(v) or nabla × v. They aren’t too difficult to understand but some visualisations are quite helpful. It you are interested look them up.
The ELI5:
The ∇ or gradient basically tells you given your current position, which direction you will travel. Imagine a ball on a hill, the direction the ball travels and how fast it travels all depend on where the ball is placed on the hill.
In the context of Maxwell’s equations: You have held a magnet before right? You have felt it push and pull things? The equations describe what a magnetic field looks like (the hill) given your position in a magnetic field, where you will be pushed or pulled (like the ball rolling).
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