Motivation behind weak form and the test function? I am not a mathematician so please bear with me. I have been reading this document to have a better understanding of FEA from the perspective of an engineer.
I am having a lot of difficulty seeing the intuition/motivation behind the weak form and why we multiply by a test function and then integrate.
Almost every text I've seen begins with declaring that we should multiply by a test function and then integrate it as if it's an axiom.
The author states that the motivation behind the weak form is the realization two vectors are equal if their inner products with some arbitrary test function is the same. Then the author continues on.
Where does this test function come from? I understand we later restrict it to be from Sobolev space for the fact that this necessitates that means the function will behave well within the domain of interest (for example by being square integrable).
Q: Why do we use the test function and integrate to find the weak form? What is the motivation/intuition? I understand everything we do afterwards, but it's the initial inception of the weak form that eludes me.
Thank you
 A: In the physical world, it is impossible to measure a function at a point. The best you can do is to measure a local average of a function over a small region near a point.
For example, we conceptualize the temperature field of the room you are sitting in as a function being defined at every point. But really, all you can do is measure the local average of the temperature near a point with a thermometer. You could manufacture more and more accurate thermometers, and the region over which you are taking the average becomes smaller and smaller. But you can never know the exact value of the temperature at a point. Temperature at a point is not even a well-defined concept, if you think about zooming in to the atomic scale.
Integrating against a test function is like taking a "measurement" of a field. By saying that
$$\int_\Omega f(x) v(x) dx = \int_\Omega g(x) v(x) dx$$
holds for all test functions $v$, we are saying that $f$ and $g$ agree for all possible measurements you could make of them. The test function $v$ is like the thermometer averaging profile from the above example.
Edit: Here's a picture I made illustrating the idea:

