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The integration Integrate[\[Delta][...]dV about the infinite sphere must evaluate to 1!

Test of your formula doesn't evaluate to one, but for the modified approach

$\delta (r-\text{r0},\theta -\text{$\theta $0},\phi -\text{$\phi $0})=\frac{\csc (\theta ) \tan (\text{$\theta $0}) \delta (\theta -\text{$\theta $0}) \delta (r-\text{r0}) \delta (\phi -\text{$\phi $0})}{r^2} $

the integration gives

sol=Integrate[ DiracDelta[r - r0]/(r^2 Sin[\[Theta]])DiracDelta[\[Phi] - \[Phi]0] DiracDelta[ \[Theta] - \[Theta]0 ]/Cot[\[Theta]0]  r^2 Cos[\[Theta]] 
, {r, 0,Infinity}, {\[Theta], -Pi/2, Pi/2}, {\[Phi], 0, 2 Pi}, 
Assumptions -> {r0 > 0, 0 < \[Phi] < 2 Pi, -Pi/2 < \[Theta] < Pi/2,Element[{r0, \[Theta]0, \[Phi]0}, Reals]} ]

(*UnitStep[\[Pi] - 2 \[Theta]0, \[Pi]/2 + \[Theta]0, 2 \[Pi] - \[Phi]0, \[Phi]0]*)

Plot3D[sol, {\[Phi]0, 0, 2 Pi}, {\[Theta]0, -Pi/2, Pi/2},AxesLabel -> {\[Phi]0, \[Theta]0} ]

The result equals 1 for 0 < \[Phi]0 < 2 Pi, -Pi/2 < \[Theta]0 < Pi/2.

addendum

The added integral ep==... contains only scalar argument inside DiracDelta[re - rp], only the Integration along r1 and r2 is affected.

It follows Integrate[...DiracDelta[re - rp],{re,0,Infinity}, {rp,0,Infinity}]= Integrate[... /.re->rp, {rp,0,Infinity}]

ep=-NIntegrate[Sin[\[Theta]e] Sin[\[Theta]p], {\[Theta]e, 0, \[Pi]}, {\[Theta]p, 0, \[Pi]} , {\[Phi]e, 0, 2 \[Pi]}, {\[Phi]p, 0,2 \[Pi]}] * 
NIntegrate[(re^2*rp^2*intE[re]*intP[rp] )/(a -b*Sqrt[intE[re]*intP[rp]]+ c*intE[re]*intP[rp]) /. re -> rp  , {rp, 0, \[Infinity]} ]
(*-0.0637158*)

The integration Integrate[\[Delta][...]dV about the infinite sphere must evaluate to 1!

Test of your formula doesn't evaluate to one, but for the modified approach

$\delta (r-\text{r0},\theta -\text{$\theta $0},\phi -\text{$\phi $0})=\frac{\csc (\theta ) \tan (\text{$\theta $0}) \delta (\theta -\text{$\theta $0}) \delta (r-\text{r0}) \delta (\phi -\text{$\phi $0})}{r^2} $

the integration gives

sol=Integrate[ DiracDelta[r - r0]/(r^2 Sin[\[Theta]])DiracDelta[\[Phi] - \[Phi]0] DiracDelta[ \[Theta] - \[Theta]0 ]/Cot[\[Theta]0]  r^2 Cos[\[Theta]] 
, {r, 0,Infinity}, {\[Theta], -Pi/2, Pi/2}, {\[Phi], 0, 2 Pi}, 
Assumptions -> {r0 > 0, 0 < \[Phi] < 2 Pi, -Pi/2 < \[Theta] < Pi/2,Element[{r0, \[Theta]0, \[Phi]0}, Reals]} ]

(*UnitStep[\[Pi] - 2 \[Theta]0, \[Pi]/2 + \[Theta]0, 2 \[Pi] - \[Phi]0, \[Phi]0]*)

Plot3D[sol, {\[Phi]0, 0, 2 Pi}, {\[Theta]0, -Pi/2, Pi/2},AxesLabel -> {\[Phi]0, \[Theta]0} ]

The result equals 1 for 0 < \[Phi]0 < 2 Pi, -Pi/2 < \[Theta]0 < Pi/2.

addendum/modified

The added integral ep==... contains   DiracDelta["re" -"rp"] which stands for the vectorform of DiracDelta (see first part of the question)

It must be substituted by

dirac = DiracDelta[re - rp]/re^2 DiracDelta[\[Phi]e - \[Phi]p] DiracDelta[\[Theta]e - \\[Theta]p]/( Sin[\[Theta]e] Cot[\[Theta]p])

Knowing Integrate[...DiracDelta[x - y],{x,0,Infinity}, {y,0,Infinity}]= Integrate[... /.x->y, {y,0,Infinity}] the integration can be partially executed. The integral can be divided into three parts

ep = -NIntegrate[ (re^2*rp^2*intE[re]*intP[rp]*1/re^2)/(a - b*Sqrt[intE[re]*intP[rp]] + c*intE[re]*intP[rp])  /.re -> rp  , {rp, 0, \[Infinity]}]*
NIntegrate[1 , {\[Phi]p, 0, 2 \[Pi]}]*
NIntegrate[1/( Sin[\[Theta]e] Cot[\[Theta]p])Sin[\[Theta]e] Sin[\[Theta]p] /. \[Theta]e -> \[Theta]p, {\[Theta]p, - Pi/2, \[Pi]/2}]    
(*-2.09081 *)

The integration Integrate[\[Delta][...]dV about the infinite sphere must evaluate to 1!

Test of your formula doesn't evaluate to one, but for the modified approach

$\delta (r-\text{r0},\theta -\text{$\theta $0},\phi -\text{$\phi $0})=\frac{\csc (\theta ) \tan (\text{$\theta $0}) \delta (\theta -\text{$\theta $0}) \delta (r-\text{r0}) \delta (\phi -\text{$\phi $0})}{r^2} $

the integration gives

sol=Integrate[ DiracDelta[r - r0]/(r^2 Sin[\[Theta]])DiracDelta[\[Phi] - \[Phi]0] DiracDelta[ \[Theta] - \[Theta]0 ]/Cot[\[Theta]0]  r^2 Cos[\[Theta]] 
, {r, 0,Infinity}, {\[Theta], -Pi/2, Pi/2}, {\[Phi], 0, 2 Pi}, 
Assumptions -> {r0 > 0, 0 < \[Phi] < 2 Pi, -Pi/2 < \[Theta] < Pi/2,Element[{r0, \[Theta]0, \[Phi]0}, Reals]} ]

(*UnitStep[\[Pi] - 2 \[Theta]0, \[Pi]/2 + \[Theta]0, 2 \[Pi] - \[Phi]0, \[Phi]0]*)

Plot3D[sol, {\[Phi]0, 0, 2 Pi}, {\[Theta]0, -Pi/2, Pi/2},AxesLabel -> {\[Phi]0, \[Theta]0} ]

The result equals 1 for 0 < \[Phi]0 < 2 Pi, -Pi/2 < \[Theta]0 < Pi/2.

The integration Integrate[\[Delta][...]dV about the infinite sphere must evaluate to 1!

Test of your formula doesn't evaluate to one, but for the modified approach

$\delta (r-\text{r0},\theta -\text{$\theta $0},\phi -\text{$\phi $0})=\frac{\csc (\theta ) \tan (\text{$\theta $0}) \delta (\theta -\text{$\theta $0}) \delta (r-\text{r0}) \delta (\phi -\text{$\phi $0})}{r^2} $

the integration gives

sol=Integrate[ DiracDelta[r - r0]/(r^2 Sin[\[Theta]])DiracDelta[\[Phi] - \[Phi]0] DiracDelta[ \[Theta] - \[Theta]0 ]/Cot[\[Theta]0]  r^2 Cos[\[Theta]] 
, {r, 0,Infinity}, {\[Theta], -Pi/2, Pi/2}, {\[Phi], 0, 2 Pi}, 
Assumptions -> {r0 > 0, 0 < \[Phi] < 2 Pi, -Pi/2 < \[Theta] < Pi/2,Element[{r0, \[Theta]0, \[Phi]0}, Reals]} ]

(*UnitStep[\[Pi] - 2 \[Theta]0, \[Pi]/2 + \[Theta]0, 2 \[Pi] - \[Phi]0, \[Phi]0]*)

Plot3D[sol, {\[Phi]0, 0, 2 Pi}, {\[Theta]0, -Pi/2, Pi/2},AxesLabel -> {\[Phi]0, \[Theta]0} ]

The result equals 1 for 0 < \[Phi]0 < 2 Pi, -Pi/2 < \[Theta]0 < Pi/2.

addendum

The added integral ep==... contains only scalar argument inside DiracDelta[re - rp], only the Integration along r1 and r2 is affected.

It follows Integrate[...DiracDelta[re - rp],{re,0,Infinity}, {rp,0,Infinity}]= Integrate[... /.re->rp, {rp,0,Infinity}]

ep=-NIntegrate[Sin[\[Theta]e] Sin[\[Theta]p], {\[Theta]e, 0, \[Pi]}, {\[Theta]p, 0, \[Pi]} , {\[Phi]e, 0, 2 \[Pi]}, {\[Phi]p, 0,2 \[Pi]}] * 
NIntegrate[(re^2*rp^2*intE[re]*intP[rp] )/(a -b*Sqrt[intE[re]*intP[rp]]+ c*intE[re]*intP[rp]) /. re -> rp  , {rp, 0, \[Infinity]} ]
(*-0.0637158*)

The integration Integrate[\[Delta][...]dV about the infinite sphere must evaluate to 1!

Test of your formula doesn't evaluate to one, but for the modified approach

$\delta (r-\text{r0},\theta -\text{$\theta $0},\phi -\text{$\phi $0})=\frac{\csc (\theta ) \tan (\text{$\theta $0}) \delta (\theta -\text{$\theta $0}) \delta (r-\text{r0}) \delta (\phi -\text{$\phi $0})}{r^2} $

the integration gives

Integrate[ DiracDelta[r - r0]/(r^2 Sin[\[Theta]])DiracDelta[\[Phi] - \[Phi]0] DiracDelta[ \[Theta] - \[Theta]0 ]/Cot[\[Theta]0]  r^2 Cos[\[Theta]] 
, {r, 0,Infinity}, {\[Theta], -Pi/2, Pi/2}, {\[Phi], 0, 2 Pi}, 
Assumptions -> {r0 > 0, 0 < \[Phi] < 2 Pi, -Pi/2 < \[Theta] < Pi/2,Element[{r0, \[Theta]0, \[Phi]0}, Reals]} ]

(*UnitStep[\[Pi] - 2 \[Theta]0, \[Pi]/2 + \[Theta]0, 2 \[Pi] - \[Phi]0, \[Phi]0]*)

The result equals 1 for 0 < \[Phi] < 2 Pi, -Pi/2 < \[Theta] < Pi/2.

The integration Integrate[\[Delta][...]dV about the infinite sphere must evaluate to 1!

Test of your formula doesn't evaluate to one, but for the modified approach

$\delta (r-\text{r0},\theta -\text{$\theta $0},\phi -\text{$\phi $0})=\frac{\csc (\theta ) \tan (\text{$\theta $0}) \delta (\theta -\text{$\theta $0}) \delta (r-\text{r0}) \delta (\phi -\text{$\phi $0})}{r^2} $

the integration gives

sol=Integrate[ DiracDelta[r - r0]/(r^2 Sin[\[Theta]])DiracDelta[\[Phi] - \[Phi]0] DiracDelta[ \[Theta] - \[Theta]0 ]/Cot[\[Theta]0]  r^2 Cos[\[Theta]] 
, {r, 0,Infinity}, {\[Theta], -Pi/2, Pi/2}, {\[Phi], 0, 2 Pi}, 
Assumptions -> {r0 > 0, 0 < \[Phi] < 2 Pi, -Pi/2 < \[Theta] < Pi/2,Element[{r0, \[Theta]0, \[Phi]0}, Reals]} ]

(*UnitStep[\[Pi] - 2 \[Theta]0, \[Pi]/2 + \[Theta]0, 2 \[Pi] - \[Phi]0, \[Phi]0]*)

Plot3D[sol, {\[Phi]0, 0, 2 Pi}, {\[Theta]0, -Pi/2, Pi/2},AxesLabel -> {\[Phi]0, \[Theta]0} ]

The result equals 1 for 0 < \[Phi]0 < 2 Pi, -Pi/2 < \[Theta]0 < Pi/2.