Figure 2 - uploaded by Thomas Osler
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... In Figure 2, the vertex is shared by the four blocks of quadrilateral . We must show that M + Q = N . Since the diagonals of parallelogram bisect each other, Since B and H are the midpoints of two sides of , and and are the midpoints of two sides of CEG , quad In the same manner, 1 4 quad . By combining these equalities we ...
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... need to determine whether it is conservative. That is, we need to determine whether F is the gradient of some potential function f . This is the second line of the diagram! We could start by checking the mixed derivatives. However, what we really want is the potential function; we should be moving up the diagram, not down. What happens if we simply integrate both components, as shown in the first drawing in Figure 2? The potential function is clearly contained in the results of these two integrals; it is just a question of combining them ...
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... attempted the same procedure, we would obtain the second drawing in Figure 2. Simply by noticing that x y , a function of two variables, only occurs once, we see that H is not conservative. We describe this to students as a murder mystery . A crime has been committed by the unknown murderer f ; your job is to find the identity of f by interviewing the witnesses. Who are the witnesses? The components of the vector field. What do they tell you? Well, you have to integrate (“interrogate”) them! Now for the fun part. If two witnesses say they saw someone with red hair, that doesn’t mean the suspect has two red hairs! So if you get the same clue more than once, you only count it once. On the other hand, some clues require corroboration. These witnesses were situ- ated in such a way that each could only look in one direction. Thus, one witness, the x -component, only sees terms involving x , etc. If a clue contains more than one variable, it should have been seen by more than one witness! In fact, functions of n variables should occur precisely n times. In the case of the vector field H , the clue x y was only seen by one witness, not both; somebody is lying! In short, clues must be consistent. Here is the Murder Mystery Method in a ...
