@naturallyInconsistent I plotted the left and right hand sides of the equation to get a feel for the solutions (which are interesting). I asked about an analytic solution to see if it happens to have one. Yes, looking for solutions near A would be of interest. In that case, allow me to pose the full problem
@TobiasFünke I was trying to calculate the best value to bid in a silent (i.e. bids are private), first price auction, assuming the goal to be maximization of the expected earnings, where "earnings" means "value of product minus amount bid in the auction."
@ACuriousMind ooooh this could be interesting. Perhaps someone with nefarious intent finds out and yes, wants to control the numbers to create outcomes beneficial to themself. But then it turns out they need quantum computation to do that... which can't be done without the random number ticker doing its thing.
What physicists should probably say is this: "This list of numbers, which describe a thing in a particular basis, represents a vector iff those numbers transform in a particular way when I change to a different basis".
It was very helpful for me, for example, to realize that a function and its Fourier transform can be viewed as different coefficients for the same vector.
@SillyGoose Sure but isn't it a lot easier to understand the basic definition of a vector as a thing that exists without any specific representation before getting into that?
The same problem happens in thermodynamics once the physics books start pushing around $dQ$, $dV$, etc. That stuff is much more clear if you focus on the maps themselves as opposed to treating $d\text{whatever}$ as some kind of pseudo-value.
When I read that, I finally realized that an $(n, k)$ tensor is a linear function that maps $n$ vectors and $k$ co-vectors to a scalar. Period. No transformation nonsense in the definition. But of course you can investigate how to represent a tensor via coefficients and investigate how those coefficients change with changing basis.
There again, they are misidentifying the values of the elements of a tensor expressed in various bases with the underlying tensor itself. it's horrible. I only understood tensors by reading Munkres's Analysis on Manifolds where the idea of an antisymmetric tensor is introduced as a function $T: V^n \rightarrow \mathbb{R}$ that maps $n$ vectors to a real number, is linear in each argument, and has the antisymmetry property that swapping any two arguments inverts the result.
This confusion oblitarates students' chances of understanding tensors because physicists write insane phrases like "A tensor is a set of numbers $g_{ij}$ that transform like a tensor".
This is most clearly evidenced when physicists write e.g. "...the function $f(x)$", which is almost nonsense. It seems what they mean is "the function whose values are $f(x)$", which as I said, is a sort of confusion between functions and values.
Consider a function $f: \mathbb{R} \rightarrow \mathbb{R}$ and a subset of the real line $X$. Then there's an integral $\int_X f$. Notice that there is no $dx$ anywhere and there's no $f(x)$ anywhere.
