Question

If f is a scalar field and F, G are vector fields, then f F, F · G, and F × G are defined by the following. (f F)(x, y, z) = f(x, y, z) F(x, y, z) (F · G)(x, y, z) = F(x, y, z) · G(x, y, z) (F × G)(x, y, z) = F(x, y, z) × G(x, y, z) Find an identical expression, assuming that the appropriate partial derivatives exist and are continuous.

Answer 1

Answer:

Step-by-step explanation:

Consider curl $(fF)$ where $f$ is a scalar function and F is a vector function

                  $F=F_{1}i +F_{2}j +F_{3}k$

                     i         j          k                

$curl(fF) = \frac{\partial}{\partial x}$      $\frac{\partial}{\partial y}$      $\frac{\partial}{\partial z}$        

                  $fF_{1}$    $fF_{2}$    $fF_{3}$  

$curl(fF)=i(\frac{\partial}{\partial y}(fF_{3})-\frac{\partial}{\partial z}(fF_{2}))-j(\frac{\partial}{\partial x}(fF_{3})-\frac{\partial}{\partial z}(fF_{1}))+k(\frac{\partial}{\partial x}(fF_{2})-\frac{\partial}{\partial y}(fF_{1}))$

$=i(\frac{\partial}{\partial y}(F_{3})+\frac{\partial}{\partial y}(f)-\frac{\partial}{\partial z}(F_{2})-\frac{\partial}{\partial z}(f))-j(\frac{\partial}{\partial x}(F_{3})+\frac{\partial}{\partial x}(f)-\frac{\partial}{\partial z}(F_{1})-\frac{\partial}{\partial z}(f))+k(\frac{\partial}{\partial x}(F_{2})+\frac{\partial}{\partial x}(f)-\frac{\partial}{\partial y}(F_{1})-\frac{\partial}{\partial y}(f))$

$curl(fF)=f(\Delta\times F)+(\Delta f)\times F$