What is the Name for the Linear Mapping f* and How is it Proven?

  • Thread starter yifli
  • Start date
In summary, the linear mapping f^*:\wedge^k(R^{m}_{f(p)}) \rightarrow \wedge^k(R^{n}_{p}) is referred to as the adjoint of f when k is 1. However, it is also known as the chain rule when k > 1. Additionally, a proof of f^*(d\omega)=d(f^*\omega) for a 0-form \omega can be shown.
  • #1
yifli
70
0
Is there a name for the linear mapping [itex]f^*:\wedge^k(R^{m}_{f(p)}) \rightarrow \wedge^k(R^{n}_{p})[/itex] where f is a differentiable mapping from [itex]R^n \rightarrow R^m[/itex].

When k is 1, f* is called the adjoint of f. But what about k > 1?

Also can someone show me a proof of [itex]f^*(d\omega)=d(f^*\omega)[/itex] where [itex]\omega[/itex] is a 0-form.

Thanks
 
Physics news on Phys.org
  • #2
i think that second one is called the chain rule.
 
  • #3
mathwonk said:
i think that second one is called the chain rule.

Thanks.

Still, can anyone show me a proof of the f*(dw)=d(f*w)
 

Related to What is the Name for the Linear Mapping f* and How is it Proven?

1. What is "Some questions about f*"?

"Some questions about f*" is a phrase used to refer to a set of inquiries or concerns related to the topic of f*. It is not a specific question, but rather a general term used to encompass various questions and discussions about f*.

2. Why is f* an important topic for scientists to study?

F* is an important topic for scientists to study because it is a fundamental aspect of insert specific field of study here. Understanding f* can help us gain a deeper understanding of the natural world and how it works. It can also lead to advancements in technology and medicine.

3. Can you give an example of a real-life application of f*?

There are countless real-life applications of f* in various fields. For example, in biology, f* can refer to the process of insert specific biological process here. In physics, f* can refer to insert specific physical phenomenon here. In both cases, understanding f* is crucial for furthering our understanding and advancements in these fields.

4. What are some current debates or controversies surrounding f*?

There are ongoing debates and controversies surrounding f* in many fields of study. For example, some scientists argue that insert specific argument here, while others argue that insert opposing argument here. These debates and controversies highlight the importance of continued research and discussion about f*.

5. How can non-scientists benefit from learning about f*?

Even for those who are not directly involved in scientific research, learning about f* can still have many benefits. It can help individuals better understand and critically evaluate scientific information and findings. Additionally, some of the advancements and discoveries made through studying f* can have practical applications in our daily lives, such as new medical treatments or technologies.

Similar threads

I Frobenius theorem for differential one forms
    • Differential Geometry
Replies
6
Views
550
I Differential operator vs one-form (covector field)
    • Differential Geometry
Replies
10
Views
731
A A claim about smooth maps between smooth manifolds
    • Differential Geometry
Replies
20
Views
2K
I Induced orientation on boundary of ##\mathbb{H}^n## in ##\mathbb{R}^n##
    • Differential Geometry
Replies
6
Views
375
I The Road to Reality - exercise on scalar product
    • Differential Geometry
Replies
2
Views
604
A Differentiability of a function between manifolds
    • Differential Geometry
Replies
3
Views
1K
I Darboux theorem for symplectic manifold
    • Differential Geometry
Replies
4
Views
2K
I What is the construction of charts for the tangent bundle on the unit circle?
    • Differential Geometry
Replies
8
Views
2K
I Understanding Integration on an Orientable Manifold
    • Differential Geometry
Replies
1
Views
1K
I Calculating the Hodge Star in the most general case
    • Differential Geometry
Replies
29
Views
1K

Hot Threads

Recent Insights

Back
Top