Dear all, I have two vector fields [itex] \vec{B}[/itex] and [itex]\vec{A}[/itex] related by: [itex] \vec{B}=\nabla \times \vec{A}[/itex] How can I simplify the following term: [itex]\frac{\partial }{\partial \vec{A}} B^{2}[/itex] where [itex]\frac{\partial }{\partial \vec{A}}=(\frac{\partial }{\partial A_{x}} \frac{\partial }{\partial A_{y}} \frac{\partial }{\partial A_{z}} )[/itex] 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).
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.