Parallelism of Time-varying Vectors

In summary, the conversation covers a solved problem with some unclear parts, specifically in parts (1), (2), and (4). The intermediate value theorem is mentioned and explained as a way to determine values of a continuous function. However, it is argued that this theorem should be formally established and referred to by name in pure mathematical proofs. Additionally, there are questions about the placement of section labels and the casual nature of assumptions in pure mathematics.
  • #1
baldbrain
236
21

Homework Statement


This is a solved problem, but I haven't understood a few things.
I've marked out sections of the solution in white for convenience. The markings are positioned where that particular section ends.
Untitled__1532764784_116.75.182.31.jpg

In part (1), how did they just assume
f1(0) = 2, f2(0) = 3, g1(0) = 3, g2(0) = 2
f1(1) = 6, f2(1) = 2, g1(1) = 2, g2(1) = 6

And, in part (4), what is this 'intermediate value theorem' that they've used?
We've just done the basics on vectors, so I have no idea where this 'intermediate value theorem' came from...
Then, in part (2) they say we have to prove that
f1(t).g2(t) - f2(t).g1(t) = 0
And in part (4), they just implied the same thing from out of nowhere & voila! The problem's over!
Please explain that too...

 

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  • #2
Your objections are valid. These values seem to come out of nowhere.

The intermediate value theorem simply says that if a continuous function has two values, then it must also have every value in between the two.
 
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  • #3
FactChecker said:
The intermediate value theorem simply says that if a continuous function has two values, then it must also have every value in between the two.
That's so obvious (assuming the function is defined on R). We've done this as a deduction of continuity, not as a separate theorem.
 
  • #4
These values have a pattern...
2 3 3 2
6 2 2 6
 
  • #5
At the beginning of the solution, it says "If A(t) ..." . So they are still defining the problem. I think that they have just put the section labels in the wrong place.

To say that the intermediate value theorem is just a "deduction of continuity" is a little too casual for some people in formal mathematical proofs. A pure mathematician would be more comfortable with this https://en.wikipedia.org/wiki/Intermediate_value_theorem#Proof . And once that theorem is established formally, they would refer to it by name. Pure math is full of casual assumptions that turned out to be wrong (thanks, Georg Cantor, you SOB).
 
  • #6
FactChecker said:
At the beginning of the solution, it says "If A(t) ..." . So they are still defining the problem. I think that they have just put the section labels in the wrong place.

To say that the intermediate value theorem is just a "deduction of continuity" is a little too casual for some people in formal mathematical proofs. A pure mathematician would be more comfortable with this https://en.wikipedia.org/wiki/Intermediate_value_theorem#Proof . And once that theorem is established formally, they would refer to it by name. Pure math is full of casual assumptions that turned out to be wrong (thanks, Georg Cantor, you SOB).
Ok professor:wink:
 

1. What is the concept of parallelism in time-varying vectors?

The concept of parallelism in time-varying vectors refers to the relationship between two or more vectors that have the same direction and magnitude at any given point in time. This means that the vectors remain parallel to each other as they change over time.

2. How is parallelism of time-varying vectors measured?

Parallelism of time-varying vectors is typically measured by calculating the dot product of the vectors at different time points. If the dot product remains constant over time, then the vectors are considered parallel.

3. What are some real-world applications of parallelism of time-varying vectors?

Some real-world applications of parallelism of time-varying vectors include analyzing movements in sports, tracking the motion of objects in video footage, and studying the behavior of dynamic systems in physics and engineering.

4. Can parallelism of time-varying vectors occur in three-dimensional space?

Yes, parallelism of time-varying vectors can occur in three-dimensional space. In fact, it is often more common in three-dimensional systems, where there are multiple planes and axes for vectors to align with.

5. How does the concept of parallelism of time-varying vectors relate to vector calculus?

In vector calculus, parallelism of time-varying vectors is a crucial concept in understanding the behavior of vector fields over time. It allows for the calculation of important properties such as line integrals, flux, and work done by a vector field.

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