Vector Analysis Identity derivation

[EngageEngage]

Homework Statement

derive the identity:
del((F)^2) = 2 F . del(F) + 2Fx (del x F)

the dot is a dot product

Homework Equations

The Attempt at a Solution

first i set F = <a,b,c>, making F^2 = a^2 + b^2 + c^2
I took the partial derivatives with respect to x, y, and z (to get the necessary parts for the gradient), which gives me:

d/dx(a^2 + b^2 + c^2) = 2a * da/dx + 2b*db/dx + 2c*db/dc
d/dy(...) = ... 2a * da/dy + 2b*db/dy + 2c*db/dy
d/dz(...) = ... 2a * da/dz + 2b*db/dz + 2c*db/dz

But, because its a gradient of a scalar, the three components above are those of a vector. After putting together the components I am not sure where to go. What actually confuses me is that in the derivation you are supposed to show that a vector equals a vector plus a scalar -- the ( 2F . del(F)) term is a scalar... . If someone could please help me out that would be greatly appreciated. Thank you

 

[Dick]

If F is a vector, then you can't interpret del(F) as a vector. It has to be a rank 2 tensor. It's ij component is del_i(F_j) or del_j(F_i). So the i component of F.del(F) would be F_j*del_i(F_j) (summed over j) or possibly F_j*del_j(F_i) (summed over k). It's kind of ambiguous. What does it mean??

 

[pabloceli]

derivation

how can you do the derivation of d = Da/2b