Articles of dimensional analysis

What is the measure of $A=\times\times\cup\times\times \setminus ^3$?

I really get stuck after one point, and don’t know where to go on.I know that my try, up to where I am stuck is correct. $$\color{#20f}{\text{TRY:}}$$ $$B_1=[-1,2]\times[0,3]\times[-2,4],\mu(B_1)=3 \cdot 3\cdot6=54 \\ B_2=[0,2]\times[1,4]\times[-1,4] \mu(B_1)=2 \cdot 3\cdot5=30$$ $\mu(A)=\mu(B_1\cup B_2)=\mu(B_1)+\mu(B_2)-\mu(B_1\cap B_2).$ and $B_1\cap B_2=([-1,2]\cap[0,2])\times([0,3]\cap[1,4])\times([-2,4]\cap[-1,4])=[0,2]\times [1,3] \times[-1,4]$ $$\mu (B_1\cap B_2)=2 \cdot 2\cdot5=20$$ therefore: $$\color{#f00}{\mu(A)=54+30-20=64.}$$ Now that I know that […]

cross-products versus units of measure

If I draw 2 perpendicular line segments on the ground, 3 meters and 4 meters, how far into the sky does their cross-product extend? What if instead the line lengths are 300 cm and 400 cm? Can someone explain cross-products to me in a way that won’t make Pythagoras angry?

Why do units (from physics) behave like numbers?

What are units (like meters $m$, seconds $s$, kilogram $kg$, …) from a mathematical point of view? I’ve made the observation that units “behave like numbers”. For example, we can divide them (as in $m/s$, which is a unit of speed), and also square them (the unit of acceleration is $\frac{m}{s^2}$). In addition to that, […]

Higher dimensional analogues of the argument principle?

I know there are higher dimensional analogues of the argument principle. (See But I do not have books about it and I cannot find anything of value on the internet for free. Plz give me references or explainations. I would like an example of how one can find the zero’s of functions in 3 […]

Arbitrarily discarding/cancelling Radians units when plugging angular speed into linear speed formula?

Why is the radians implicitly cancelled? Somehow, the feet just trumps the numerator unit. For all other cases, you need to introduce the unit conversion fraction, and cancel explicitly. Is it because radians and angles have no relevance to linear speed (v), so they are simply discarded?

What mathematical structure models arithmetic with physical units?

In physics we deal with quantities which have a magnitude and a unit type, such as 4m, 9.8 m/s², and so forth. We might represent these as elements of $\Bbb R\times \Bbb Q^n$ (where there are $n$ different fundamental units), with $(4, \langle1,0,0\rangle)$ representing 4m, $(9.8, \langle1,0,-2\rangle)$ representing 9.8 m s⁻², and so forth, with […]