Differences between Del Operators at Field/Source Points

[newbie101]

Hello All,

May I know what is the difference between
1) Del operator with respect for field point
2) Del operator with respect to source point

thanks
newbie

 

[jdavel]

newbie,

Not sure what you mean.

Del is an operation on a scalar that gives a vector (namely, the gradient of the scalar)

What are the "field" and "source" that you're talking about?

 

[arildno]

I am quite certain that he is confused by the convention of regarding the divergence of a source potential as a multiple of dirac's delta function.

However, only newbie knows for sure..

 

[newbie101]

Hi All,

thanks for helping.. let me explain
i'm reading this text on the derivation of helmholtz theorem

let me just quote directly from the book

Page 2 top half
" In Equations (A-2) through (A-5), the operator 'del-f' differentiates with respect to field point rf, while the operator 'del-s' differentiates with repect to the source point rs"

May I know the difference between the operators here.


Page 2 bottom half
" From Equation (A-1) since F(rs) is a function of the source point alone, but "del-f" differentiates with respect to the field point... "

Well apparently we can move F(rs) out of the lapacian here. Please help explain how this is possible

thanks again
newbie101

 

[jtbell]

page 2 top half
" In Equations (A-2) through (A-5), the operator 'del-f' differentiates with respect to field point rf, while the operator 'del-s' differentiates with repect to the source point rs"

That means that, for example,

$$\nabla_f V = \frac {\partial V}{\partial x_f} \hat {\bold i} + \frac {\partial V}{\partial y_f} \hat {\bold j} + \frac {\partial V}{\partial z_f} \hat {\bold k}$$

whereas

$$\nabla_s V = \frac {\partial V}{\partial x_s} \hat {\bold i} + \frac {\partial V}{\partial y_s} \hat {\bold j} + \frac {\partial V}{\partial z_s} \hat {\bold k}$$

where V is some function of $x_f$, $y_f$, $z_f$, $x_s$, $y_s$, and $z_s$ (that is, depends on both the field coordinates and the source coordiates).

 

[jdavel]

jtbell,

So your vector V is analogous to the Green's function G(rs,rf) since it's a function of both rs and rf. But since F(rs) is a function only of rs, it doesn't vary with rf, so when derivatives are taken wrt rf, F acts like a constant.

newbie, does that help at all, or am I missing your point entirely?

 

[newbie101]

jtbell & jdavel,

yes it does explain everything if vector V here is a function of both (x,y,z) at field point and (x,y,z) at source point... which should be the case

since the E field at a point would depend on both
1) where the field point is as well as
2) where the source is


however, I am still not understanding the partial derivative here ... i mean how is dV/dXf different from dV/dXs ... arent there only 3 axis here X,Y,Z so the gradient whould still be the same wouldn't it ?

thanks again all
newbie101

** if necessary, i can scan more pages **

BTW the book is "Numerical Computation of Electric and Magnetic Fields" by Charles W Stelle

 

[jdavel]

since the E field at a point would depend on both
1) where the field point is as well as
2) where the source is

newbie, When you say "the E field at a point would depend on...where the source is" it sounds like you think the source is located at a single point. That's not true here; the source is distributed over the entire volume.

 

[newbie101]

Yes the source is distributed. Thanks jdavel