I am confused. I never seen derivative of a vector respect to another vector. When I go on the web, the article just show divergence, curl, gradient etc. But not derivative of a vector respect to another vector?
For example what is For is a constant vector.
The book seems to imply:
I guess I don't know how to do a derivative like this. Can anyone help?
Assuming that we are talking about vectors in [tex]R^[tex] (more specifically ) and not abstract vector spaces, my interpretation of would be a "scalar product"- the square of its length. In that case, this is a function from a vector space to its underlying field (probably the real numbers). Assuming this is a 3 dimensional vector space, we can think of that as a function of three variables, the x, y, z, components of the vector and then the "deriviative of " is the "gradient" vector, . If we have, specifically, then the derivative, or gradient, is as claimed.
(Strictly speaking, the derivative, at a given , is the linear function that maps to but we can think of that as the dot product of with just as we think of the derivative of f(x) at to be the number when, more deeply, it is the linear function .
If you have an actual "vector" function from to , say, , then we can think of its derivative as the matrix
.
More correctly, it is the linear transformation defined by that matrix.
Sorry guys and thanks for the help.
I type wrong, it was , sorry.
The book is PDE by Strauss p194 to p195. It is part of the derivation of the Green's Function for sphere. The part is about normal derivative of G. It talked about derivation respect to and some very funcky statement I still don't understand. But the later part just went back to the ordinary definition of normal derivative:
and derive the equation accordinary as if nothing happened!!!! So it is a non question at this point. Strauss is not a good book in any stretch. I just cannot find any PDE book that cover the Green's Function and the EM book that I ordered is still in shipment!!!
Thanks
Alan