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Vector Analysis: Help needed 
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#1
Jul312, 12:35 PM

P: 409

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. 


#2
Jul412, 04:17 AM

P: 4,572

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). 


#3
Jul412, 07:48 AM

P: 409

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. 


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