Simplifying Vector Analysis Derivatives with \nabla \times \vec{A}: Expert Tips

[Hassan2]

Dear all,

I have two vector fields $\vec{B}$ and $\vec{A}$ related by:

$\vec{B}=\nabla \times \vec{A}$

How can I simplify the following term:

$\frac{\partial }{\partial \vec{A}} B^{2}$

where $\frac{\partial }{\partial \vec{A}}=(\frac{\partial }{\partial A_{x}} \frac{\partial }{\partial A_{y}} \frac{\partial }{\partial A_{z}} )$

I would also like to know what are this kind of derivatives ( derivatives with respect to a vector field) called.

Thanks.

 

[chiro]

Dear all,

I have two vector fields $\vec{B}$ and $\vec{A}$ related by:

$\vec{B}=\nabla \times \vec{A}$

How can I simplify the following term:

$\frac{\partial }{\partial \vec{A}} B^{2}$

where $\frac{\partial }{\partial \vec{A}}=(\frac{\partial }{\partial A_{x}} \frac{\partial }{\partial A_{y}} \frac{\partial }{\partial A_{z}} )$

I would also like to know what are this kind of derivatives ( derivatives with respect to a vector field) called.

Thanks.

Hey Hassan2.

Try expanding out the cross product of del and A first.

Also when you say the vector derivative, are the elements of each vector mapped to the same corresponding element in the other? In other words if A = [x0,y0,z0] and B = [x1,y1,z1] then is x0 = f(x1), y0 = g(y1) and z0 = h(z1) (and the components are completely orthogonal)?

If this is the case, you will be able to expand del X A using the determinant formulation and simplify terms depending on how you define your elements of your vector (even if they are more general than above).

 

[Hassan2]

The elements of the vectors are NOT mapped correspondingly. In fact the first equation is the definition of B, thus, the components are intertwined.

I couldn't simplify it by expanding the curl.It results in partial derivatives of second order multiplied by partial derivatives of first order.

Thanks.