Answer to Question #260107 in Physics for OMAR

Question

How do I know in which law I use this symbol ≡.

(the triple sign).

Please give me 10 uses for the symbol c.

I will tell you my eight uses of the symbol ≡ and please tell me whether they are right or not, then give me other examples about how to use it.

 I use ≡ like:

1- (absolute value of A) ≡ lAl = A

2- 3≡3

3- T₁=T₂≡T

4- 3+1 = 2+2 ≡ 4

5- Also in FUNCTIONS:

f ≡g it means that the graphs of f and g are the same.

6- sin²θ + cos²θ ≡ 1

7- a ≡ m/s²

8- W≡V.A

Is the above true?

Give me more examples if it is true or wrong.

also there is another issue I want to ask you about.

How to use:

$\intop$ and $\iint$ and $\iiint$

and what are the differences between:

$\intop$ $\iint$ $\iiint$ and $\oint\oiint\oiiint$

Answer 1

Q: How do I know in which law I use this symbol ≡?

A: If it is a strictly and explicitly (obviously) defined equation without any hidden contents, like x=2t, use =, otherwise, use ≡. Example: if it is something equivalent to something, like "a is proportional to b", you can write a=kb (strictly, quantitatively, obviously, you know the way, you know how a is proportional to b: just multiply by k) or a≡b (qualitatively, not obvious, this form says the reader "Hey you see a is proportional b don't bother yourself how exactly it is proportional just know it is").

My personal advice: Try to use ≡ less often, it is mostly for quite complex math, not physics.

1 - Correct

2 - 3 is strictly and always equals to 3, so, 3=3 is correct, 3≡3 is wrong.

3 - Correct, although T₁=T₂=T is also true. Try to use ≡ less often, it is mostly for math, not physics.

4 - Wrong, 3+1 = 2+2 = 4 [it is always 4 and nothing else, there's no hidden meaning in 3+1 = 2+2 = 4, use =].

5 - Correct because you speak about graphs, visible things, not quantitative things, two graphs can look the same but be defined absolutely differently, so here you can use the triple sign.

6 - The sum of sine squared and cosine squared is always 1, it's explicit, no hidden sense, use =.

7 - Correct because quantitatively a is not equal to meters per second squared, so you can't use = but should use ≡

8 - Same as 7, correct.

In the previous answers, we gave you good examples, all things I can imagine now are no new, so, hope, this explains well. The only case we didn't mention is this, it's from number theory and is used in cryptography (very complex math):

$$x^n\equiv q(\text{mod } n).$$

As you see, it's not explicit. It's not obvious. There's a lot of hidden sense in this equation, that's why there's ≡.

$\intop$ and $\iint$ and $\iiint$ - single, double, triple integral (depends on how many variables you have).

$\intop$ is a regular integral and ∮ is line integral (closed curve/line/contour integral), the latest implies you integrate something over a loop. The simplest example: such integration along a circle will give you the circumference (length of the circle).