Before reading this question, let me just state that prior to this year, I had allways thought vectors were arrows in space or lists of numbers, and I am still getting accustomed to their more formal definition of meeting a list of rules in their operations, having a magnitude, and some "direction" (although that is not necessarilly a direction in physical space).
My question is this: Are all derivatives vectors? Even of things that aren't?
Allow me to explain:
I was reading in a physics book that although angular displacement in itself is not a vector, angular velocity and acceleration is.
This makes sense: Consider 2 rotations on an object, X and Y.
$X$ spins the object around the X-axis and $Y$ spins the object around the Y-axis.
Saying that we are going to produce $X+Y$ means we are going to spin the object around the X-axis and then the Y-axis while saying we are going to do $Y+X$ means we are going to rotate the object around the Y-axis and then around the X-axis.
As you can see from the picture below, $X+Y\neq Y+X$
However, as the change becomes infinitely small, the prospect of applying $Y$ after $X$ or $X$ after $Y$ becomes meaningless, since there is an infinitely small amount of time between the application of each tiny rotation, meaning that $dX/dt + dY/dt = dX/dt + dX/dt$. It doesn't matter whether we apply an infinitely small rotation in the X direction first or in the Y direction first, the rotations are infinitely small, so applying one after the other kind of loses all meaning. They are being applied at the same time.
We can also justify other vector properties via the same reasoning, such as $C(w_{X} + w_{y}) = Cw_{X} + Cw_{y}$ ($w$ is the symbol for rotational velocity). While with overall rotations, multiplying by the constant made a difference in the final position, since the velocities are over infinitely small time intervals and are therefore being applied at the same time, scaling each of the velocities by a constant is the same thing as applying each of the scaled velocities one after the other (which has lost all meaning when we are referring to derivatives) and scaling them separately: it doesn't make a difference their "magnitude" relative to the other angular velocity, since at the end of the day, since we are considering infinitely small time intervals, they will still be happenning at the same time, unlike overall rotations which will be happenning one after the other and making one of the rotations greater may affect the total effect after applying the other.
We can even think of more examples where this applies, such as (and this is a completely random example just to get my point across) the effects of temperature change, let's say. While raising the temperature to 1000 degrees and then lowering it to -500 will leave you as frozen ashes, but lowering to -500 and then raising it to 1000 will leave you with ashes surrounded by steam, doing each of them in infinitely small intervals, it doesn't matter if you start by raising the temperature or start by lowering it, at the end of the day you will have the same result (if the temperature is being raised faster than it is being lowered, the room will get hotter, and if it's being lowered faster than its being raised, the room will get colder, and if they're happening at the same time you won't end up as ashes OR frozen since the temperature won't change at all).
Anyways, are there other examples such as this, where the derivative is a vector but the change in itself is not? Are all derivatives vectors?
Thanks!
I regard it as a kind of convenient coincidence that we can use a vector parallel to the axis of rotation (the direction in which there is no movement!) in order to represent a rotational rate in a physics problem.
A vector space can have any number of dimensions. It could have two dimensions, for example. Rotations are possible in two dimensions, but there isn't any obvious choice of a two-dimensional vector corresponding to any given rotational rate. We need only one number to measure the rotational rate; if we used a two-dimensional vector, one of the dimensions would be redundant.
A vector space can have four dimensions. Rotations are possible in four dimensions too. But a single four-dimensional vector is not adequate to describe a rate of rotation. Rotational rates in four dimensions have six parameters.
It's just in a three-dimensional vector space that the number of parameters required to describe a rotational rate happens to be the same as the number of parameters provided by a vector in the same vector space. That's what I mean by a "convenient coincidence."
On the other hand, all kinds of things can be vectors. They don't have to have an obvious "spatial direction" interpretation. The rotational rates of objects in a two-dimensional space can be considered as vectors in a one-dimensional vector space, and the rotational rates of four-dimensional objects can be considered as vectors in a six-dimensional vector space.
The real numbers form a vector space, so if you have one machine cooling a room at the rate of $-500$ and another machine heating the room at the rate of $1000,$ you can add the derivatives to get the result $500,$ either as a vector sum or a sum of numbers (the same thing in this case).
All you need is objects that obey the rules that define a vector space, and the desire to interpret those objects as some kind of vector (possibly a very abstract kind of vector); then you can say those objects are vectors (of some sort).