Extracting Vector F: A General Question

In summary, the conversation discusses the possibility of extracting a vector F from the equation curl F = B, where both F and B are vectors and B is a constant vector. It is mentioned that a solution is only possible if B is perpendicular to the vector \nabla, which is represented by div B = \nabla \cdot B = 0. However, there is no unique solution as there are many possible vector fields with curl zero. One suggestion is to try using linear functions of the coordinates to extract F.
  • #1
24
0
Hi everybody ...


I have a general question ..


if i have for example...

curl F = B

F and B are vectors and B is constant vector ..

is there any way to extract vector F?

thanks
 
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  • #2
Must [tex]F[/tex] be a vector field [tex]F = F_{x}\widehat{i} + F_{y}\widehat{j} + F_{z}\widehat{k}+...[/tex]? I thought [tex]\nabla \times F[/tex] only operates if [tex]F[/tex] is a vector field.
 
  • #3
Yes, that's true. He said "vector F" several times.

[tex]\nabla\times F= \left(\frac{\partial F_y}{\partial z}- \frac{\partial F_z}{\partial y}\right)\vec{i}- \left(\frac{\partial F_x}{\partial z}- \frac{\partial F_z}{\partial x}\right)\vec{j}+ \left(\frac{\partial F_y}{\partial x}- \frac{\partial F_x}{\partial y}\right)\vec{k}[/tex]
so that would be essentially solving the system of equations
[tex]\frac{\partial F_y}{\partial z}- \frac{\partial F_z}{\partial y}= B_x[/tex]
[tex]\frac{\partial F_x}{\partial z}- \frac{\partial F_z}{\partial x}= B_y[/tex]
[tex]\frac{\partial F_y}{\partial x}- \frac{\partial F_x}{\partial y}= B_z[/tex]

Since we can think of the cross product of two vectors as giving a vector perpendicular to both, that system, and the original equation, has a solution only if B is "perpendicular" to the "vector" [itex]\nabla[/itex]", that is if [itex]div B= \nabla\cdot B= 0[/itex].
 
  • #4
M.M.M said:
Hi everybody ...


I have a general question ..


if i have for example...

curl F = B

F and B are vectors and B is constant vector ..

is there any way to extract vector F?

thanks

There's no unique solution since there are a lot of vector fields with curl zero. If you just want anyone try messing around with linear functions of the coordinates, like F=(0,a*x,b*x+c*y).
 

What is the purpose of extracting Vector F?

The purpose of extracting Vector F is to identify and isolate this specific vector from a larger dataset for further analysis or use in a specific experiment or research project.

What techniques are commonly used for extracting Vector F?

Common techniques used for extracting Vector F include cloning, PCR, and gel electrophoresis. These methods involve manipulating DNA and using enzymes to cut and separate specific segments.

What are some applications of Vector F in scientific research?

Vector F can be used in a variety of research applications, such as creating a genetically modified organism, studying gene expression, or producing recombinant proteins for medical or industrial purposes.

What are some challenges or limitations in extracting Vector F?

One challenge in extracting Vector F is ensuring the purity and accuracy of the extracted vector, as contamination or errors in the extraction process can affect the results of downstream experiments. Another limitation is the size and complexity of the vector, which may require specialized techniques or equipment for extraction.

How can the extraction of Vector F contribute to scientific advancements?

The extraction of Vector F can contribute to scientific advancements by providing a specific tool or component for research, allowing scientists to study and manipulate genetic material in a controlled and targeted manner. This can lead to new discoveries and insights in various fields of study.

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