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Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach
Grad, curl, and div are essentially all the same operation: the exterior derivative.
Grad takes a scalarfield and gives you a vectorfield.
Curl takes a vectorfield and gives you a vectorfield.
Div takes a vectorfield and gives you a scalarfield.
But there's more to it. If you look at the resulting fields from these operations and then perform a change of variables (in manifold speak, you view the same fields in a different chart), they don't look right.
It turns out there is another kind of object called a k-form. These give to every point in space an alternating k-linear form on the tangent space. In other words, you give it k tangent vectors and it will spit out a number. The adjective alternating means if any tangent vector is repeated, the number output is zero. And the number is multilinear (ie: it's linear in each input separately, so doubling the length of any tangent vector doubles the number it spits out).
The 0-forms of a manifold are just scalarfields. The 1-forms are covectorfields. (You give a tangent vector and it spits out a number). As geometric objects, 1-forms look exactly like vectorfields, but they act different under a change of coordinates. You might say they transform "correctly".
A 1-form can be integrated over a curve. The result is a line integral, just like in usual vector calculus. However, because our objects now transform correctly, changing coordinates works as it should.
So grad, rather than being a mapping from scalarfields to vectorfields is actually a mapping from 0-forms (also just scalarfields) to 1-forms (covectorfields).
Similarly, curl isn't a mapping from vectorfields to vetorfields. Instead, it is a mapping from 1-forms to 2-forms.
A 2-form takes two tangent vectors and spits out a number. Intuitively, it returns the signed area spanned by the two tangent vectors.
Note that even though both 1-forms and 2-forms naively sync up with the notion of "vectorfield", they both act differently than a vectorfield and from each other. Under the hood, it has to do with the properties of the wedge product, written ∧, the basic operation for combining forms.
In R^3, we have standard basis x, y, and z. Well, it turns out that the k-forms also have a standard basis. For 1-forms, our basis is dx, dy, and dz. (The covector dx eats a vector and tells you what its x-component was, etc). For 2-forms, we just wedge things together: dx ∧ dy, dy ∧ dz, and dz ∧ dx form a basis. (One of our properties for wedge products is that a ∧ b = -(b ∧ a), so dy ∧ dx wouldn't be included in the basis if dx ∧ dy was). Meanwhile, 3-forms have a basis consisting of just of the triple-wedge dx ∧ dy ∧ dz.
Count the dimension of these spaces. At any point in our manifold, the 1-forms form a 3-dimensional vectorspace and the 2-forms also form a 3-dimenisonal vectorspace... but they have different bases. The 3-forms are 1-dimensional and (trivially) the 0-forms are also 1-dimensional. But again, they are not the same space!
So finally, div is a mapping from 2-forms to 3-forms.
The k-forms are admittedly very convoluted and tricky to work with. They are hardly intuitive compared to what you learn in vector calculus. But they have the advantage of playing nicely with change of coordinates. Maybe more importantly, they also generalize to any number of dimensions. Grad, curl, and div only really work in R^3. But the theory of electromagnetism and the theory of relativity take place in R^4.
For a nice introduction to the subject, you might want to check out Hubbard and Hubbard's excellent and pragmatic introduction to the subject.
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
There's this super famous book by Hubbard&Hubbard called Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach that supposedly makes forms super easy and intuitive. I have studied the LinAl parts of it and they were good. I didn't get to forms yet, but when I need to know about forms I'll be studying out of it.
https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0136574467
Plus you get to learn differential forms!