What is the purpose of using the nabla operator in this equation?

[curupira]

A simple question:
In a homework I find :
F1 X nabla X F2 where X is the simbol of cross product

I know that AX(BXC)= (A*C)*B-(A*B)C

Where* here is used to divergence

In the next step it was:

-Nabla*(F2)F1 + nabla(F1*F2)

I don't understant it, why?

 

[curupira]

The basic idea is What the diference F*nabla and Nabla*F

 

[DrGreg]

The basic idea is What the diference F*nabla and Nabla*F

[itex](\textbf{F} \cdot \nabla) \textbf{G}[/itex] is a vector whose ith component is

[tex]\left[ (\textbf{F} \cdot \nabla) \textbf{G} \right]_i = \sum_j F_j \frac{\partial G_i}{\partial x_j} [/tex]​

whereas

[tex]\left[ (\nabla \cdot \textbf{F}) \textbf{G} \right]_i = \sum_j \frac{\partial F_j}{\partial x_j} G_i [/tex]​

 

[Danger]

Jeez, man! Will you please stop posting homework questions in the wrong places?

 

[paranormal]

The nabla operator, represented by the symbol ∇, is a mathematical tool used in vector calculus to represent the gradient or rate of change of a function. In the context of the equation you provided, the nabla operator is being used to calculate the cross product of two vector fields, F1 and F2. This allows us to find the direction and magnitude of the vector resulting from the cross product.

In the next step, the nabla operator is being used to calculate the divergence of the vector field F2, which represents the net flow of the vector field from a given point. This is then multiplied by the vector field F1, which results in a vector quantity. The second term, nabla(F1*F2), represents the dot product of the two vector fields, which gives us a scalar quantity.

Overall, the purpose of using the nabla operator in this equation is to manipulate vector fields and calculate their properties, such as gradient, divergence, and curl. It is a useful tool for solving problems in physics, engineering, and other scientific fields.