Calculating when a covector is zero.

  • Thread starter Kreizhn
  • Start date
  • Tags
    Zero
In summary, the points where df_p=0 are those where ai(p)=0 for all i=1,...,n, as the differential df_p is the zero map in those cases.
  • #1
Kreizhn
743
1

Homework Statement


Hopefully an easy question. Let M be an n-dimensional smooth manifold and [itex] f: M \to \mathbb R [/itex] a smooth function. Assume the covector field df is given in some coordinate basis as
[tex] df = \sum_{i=1}^n a_i dx^i [/tex].
Find all points [itex] p \in M [/itex] such that the [itex] df_p = 0 [/itex]

The Attempt at a Solution


I'm pretty sure all that needs to be done is find all point [itex] p \in M [/itex] such that [itex] a_i(p) = 0 [/itex] for all i=1,..n. However, I just want to make sure nothing tricky is going on here that I'm missing. My reasoning is that if there is an [itex] a_i(p) \neq 0 [/itex] then
[tex] a_i(p) dx^i \left(\left. \frac{\partial }{\partial x^i}\right|_p \right) = a^i(p) \neq 0 [/tex]

Are there other tricky cases that I might not be seeing? I can imagine that for fixed vector fields there are points where the differential could yield zero, but I take the statement [itex] df_p = 0 [/itex] to mean that the 1-form is identically the zero map at that point.
 
Physics news on Phys.org
  • #2


Your reasoning is correct. The only points where df_p=0 are those where ai(p)=0 for all i=1,...,n. This is because the differential df_p is a linear map from the tangent space at p to the real numbers, and if it is identically the zero map, then it must map every tangent vector to 0. Therefore, the covector field df must have all coefficients ai equal to 0 at that point.
 

1. What is a covector and why is it important in calculations?

A covector is a mathematical object that represents a linear map from a vector space to its underlying field. It is important in calculations because it allows for the representation of quantities such as gradients and differentials, which are essential in many fields of science and engineering.

2. How do you calculate when a covector is zero?

To calculate when a covector is zero, you need to use the definition of a covector as a linear map. This means that the covector will be zero when all of its components are equal to zero. You can then use this information to solve for the variables in the covector equation.

3. Can a covector ever be zero?

Yes, a covector can be zero. This occurs when all of its components are equal to zero, as mentioned in the previous answer. However, it is important to note that not all covectors will be zero, as they can have non-zero components as well.

4. What implications does a covector being zero have in calculations?

If a covector is zero, it means that the linear map it represents is not able to differentiate between different points in the vector space. This can have implications in various calculations, such as determining the direction and magnitude of a gradient or the rate of change of a function at a specific point.

5. How can I use the concept of a covector being zero in real-world applications?

The concept of a covector being zero is used in a wide range of real-world applications, including physics, engineering, and economics. For example, in physics, covectors are used to represent forces and energy, and understanding when a covector is zero can help determine equilibrium states. In economics, covectors are used to represent utility and demand, and knowing when a covector is zero can help find optimal solutions in decision-making processes.

Similar threads

  • Calculus and Beyond Homework Help
Replies
24
Views
671
  • Calculus and Beyond Homework Help
Replies
3
Views
112
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
200
  • Calculus and Beyond Homework Help
Replies
2
Views
169
  • Differential Geometry
Replies
10
Views
660
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
560
  • Calculus and Beyond Homework Help
Replies
9
Views
762
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Back
Top