newtonian mechanics - Angular velocity of precession

So in my textbook they say this ${\rm d}\theta$ = |$d\vec{L}$|/|$\vec{L}$|

$d\vec{L}$ is the change in angular momentum caused by a torque whose vector is perpendicular to $\vec{L}$, which is the angular momentum vector of a spinning disk. Think of a gyroscope.

Imagine a right-angle triangle whose vertical component is $d\vec{L}$ and horizontal component is $\vec{L}$. Next to the hypotenuse and the horizontal component is angle $\theta$.

I assume they say ${\rm d}\theta$ = |$d\vec{L}$|/|$\vec{L}$|

because d$\theta$ is so small and close to zero. BUT HOW IS IT THAT IT IS EQUAL? I mean I can see they are roughly equal but why are they equivalent?

I can't find a proof anywhere.

If anything I'm talking about the angular velocity of precession. That is, how the original axis of angular momentum is changing with respect to time, due to some torque acting on the system.

I kind of feel we're treating differentials too freely in physics. But it is true that I only know first-year calculus so I don't know anything about differentials.

Thank you.

Answer

This is an extended comment on a side issue you mentioned. Physicists are often much sloppier than mathematicians about mathematical rigor. There are good reasons for this.

Mathematicians model ideas like vectors - things that can be added together and multiplied by numbers. Guided only by logic, they find properties that are always always true. For example, every vector space has a dimension. They have to cover oddball cases that physicists can mostly ignore. A vector space can have 0 dimensions or infinite dimensions.

Rigor is absolutely essential. One small contradiction makes it logically possible to prove anything. The whole structure of mathematics collapses. This problem has literally held up mathematics for thousands of years. Archimedes discovered the ideas of calculus. But the ancient Greeks found contradictions and paradoxes in infinity and irrational numbers. They could not come up with trustworthy answers. So they fell back on geometry. Euclid had brought rigor to geometry. And for almost 2000 years, mathematics was largely limited to geometry.

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Physicists model the behavior of the universe. Since Galileo, they have used mathematics as a tool for this purpose. To use a more modern example, forces add together the same way that vectors do.

Physicists don't need absolute rigor. They have experiments to tell them when answers are wrong. Calculations in physics are often so complicated that they cannot be done. Approximations are better than nothing, and often are quite good.

Physicists are willing to take shortcuts that are mathematical nonsense. For example, the step function of height 1 is discontinuous. It has no derivative at the step. Physicists take the derivative anyway. It is the Dirac delta function. The value is 0 everywhere except one point. The value at that point is infinite.

If you integrate the Dirac delta function you must get back the step function. This means the area under the infinite point is 1.

This is not as insane as it sounds. A functions that is almost a step, but smoothly changes from 0 to 1, does have a derivative. The derivative has a tall narrow peak. Physicists just take this idea to an extreme because it turns out to be very useful.