Understanding Vector Theory Proofs: Solutions to Common Questions

In summary: in summary, the notes attached here: https://www.physicsforums.com/showthread.php?p=3042019#post3042019 say that:- a) eqn 27 is from (\frac{\partial}{\partial x^\mu})_p (f) = \frac{\partial f}{\partial x^\mu})_p - b) eqn 29 is from (\frac{\partial}{\partial x^\mu})_p (f)- c) eqn 25 is from (\frac{\partial}{\partial x^\mu})_p (f) = (\frac{\partial f}{\partial x^\mu})_p
  • #1
latentcorpse
1,444
0
In the notes attached here:
https://www.physicsforums.com/showthread.php?p=3042019#post3042019
(apparently I can't attach the same thing in multiple threads?)
I have quite a few problems with one of the proofs. In the proof of the proposition on p15,

a) he says to note that [itex]\nu(0)=0[/itex]. why is this?

b) he goes from
[itex] \{ \frac{d}{dt} [ \alpha ( x^\mu ( \lambda(t)) - x^\mu(p)) + \beta (x^\mu(\kapa(t))-x^\mu(p)) + x^\mu(p)] \}_{t=0} = [ \alpha ( \frac{d x^\mu ( \lambda (t))}{dt})_{t=0} + \beta ( \frac{dx^\mu ( \kappa ( t))}{dt} )_{t=0}][/itex]
I really don't understand how these two lines are equal at all!
And also how can we change the [itex]\phi[/itex]'s to [itex]x^\mu[/itex]'s in going from eqn 25 to the defn of [itex]Z_p(f)[/itex]?

c) where does eqn 27 come from? isn't [itex]( \frac{\partial}{\partial x^\mu})_p (f) = \frac{\partial f}{\partial x^\mu})_p[/itex]
is it something like if we compose the numerator with [itex]\phi^{-1}[/itex] then we have to cancel that out by composing the [itex]p[/itex] with [itex]\phi[/itex] to give the [itex]\phi(p)[/itex]? I don't really get why this is allowed though?

d)Where does eqn 29 come from?

Thanks a lot for any help. I really need to get my head round all this vector business over the holidays!
 
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  • #2
For a, [tex]\lambda(0)=\kappa(0)=p[/tex], so once you plug those in you get
[tex] \nu(0)=\phi^{-1}(\alpha(\phi(p)-\phi(p)) + \beta(\phi(p)-\phi(p) +\phi(p)) = \phi^{-1}(\phi(p)) = p[/tex] (not 0 like you say in your post)

For (b), just distribute the derivative linearly. Then [tex] \frac{d x^{\mu}(p)}{dt} = 0[/tex] because p is just a fixed point, so that's just the derivative of a number
 
  • #3
Office_Shredder said:
For a, [tex]\lambda(0)=\kappa(0)=p[/tex], so once you plug those in you get
[tex] \nu(0)=\phi^{-1}(\alpha(\phi(p)-\phi(p)) + \beta(\phi(p)-\phi(p) +\phi(p)) = \phi^{-1}(\phi(p)) = p[/tex] (not 0 like you say in your post)

For (b), just distribute the derivative linearly. Then [tex] \frac{d x^{\mu}(p)}{dt} = 0[/tex] because p is just a fixed point, so that's just the derivative of a number

hi there. thanks for your answers.

do you have any ideas for c) or d) or also, how we get teh formula for [itex]Z-p(f)[/itex] in the first place?

thanks!
 

1. What is vector theory and why is it important?

Vector theory is a mathematical concept that involves the use of vectors, which are quantities that have both magnitude and direction. It is important because it provides a way to represent and manipulate physical quantities that have both magnitude and direction, such as force, velocity, and acceleration.

2. What are vector proofs and why are they challenging?

Vector proofs are mathematical demonstrations that use vector theory to solve problems and prove theories. They can be challenging because they require a deep understanding of vector concepts and their applications, as well as strong analytical and critical thinking skills.

3. How can I improve my understanding of vector theory proofs?

To improve your understanding of vector theory proofs, it is important to first have a strong foundation in vector theory. This can be achieved through studying and practicing basic vector operations, such as addition, subtraction, and scalar multiplication. In addition, familiarizing yourself with common vector theorems and their applications can also help. Practice solving a variety of vector problems and seek help from textbooks, online resources, or a tutor if needed.

4. What are some common mistakes to avoid when solving vector theory proofs?

One common mistake when solving vector theory proofs is not paying attention to the direction of vectors. Remember that vectors have both magnitude and direction, so it is important to consider both when manipulating them. Another mistake is not properly applying vector operations, such as using the wrong formula or forgetting to convert units. It is also important to double check your calculations and solutions to avoid simple errors.

5. How can I apply vector theory proofs in real life?

Vector theory has many real-life applications, particularly in the fields of physics, engineering, and computer graphics. For example, understanding vector theory can help you calculate the trajectory of a projectile, design a bridge that can withstand different forces, or create realistic 3D computer graphics. Additionally, many everyday tasks, such as navigation using maps or GPS, also rely on vector theory principles.

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