### 147

**APPENDIX C **

### We want to show that:

### , _{m} _{m} _{s} _{m} .

_{m}

_{m}

_{s}

_{m}

*S* *S*

*dS* *dS*

### φ ξ φ ξ φ ξ

### ∇ = ∇ ⋅ ∫∫ = − ∫∫ ∇ ⋅ ^{ } (C.1) In order to do that, it is convenient to choose a Cartesian reference *system with the x and y axis belonging to the plane defined by the generic * *face S of the triangle domain and with the origin in one of the triangle * vertex (Fig. C.1).

**Figure C.1. Cartesian reference system belonging to the triangle face S. **

**Figure C.1. Cartesian reference system belonging to the triangle face S.**

### Noting that with the chosen reference system the RWG basis does not *present any z-component, we obtain: *

### ( ) ( )

### ( ) , ( ) ( ) , ( )

### , , , ,

### .

### , , , , , ,

*m* *m*

*S*

*m x* *m y*

*S*

*x y z* *x y dxdy*

*x y z* *x y* *x y z* *x y* *dxdy*

*x* *y*

### φ ξ φ ξ

### φ ξ φ ξ

### ∇ = ∇ ⋅ =

### ∂ + ∂

### ∂ ∂

###

## ∫∫

## ∫∫ ^{(C.2) }

### By using the definition (2.20), the (C.2) can be written as:

*VECTOR IDENTITY *

### 148

### ( ) ( )

### , , , , , .

### 2 2

*m* *m*

*m*

*m* *m*

*S* *S*

*l x* *l x*

*x y z* *dxdy* *x y z* *dxdy*

*x* *A* *y* *A*

### φ ξ φ φ

±

### ±

±### ±

### ∂ ∂

### ∇ = ± ±

### ∂ ∂

## ∫∫ ∫∫ ^{(C.3) }

### The first integral in (C.3) can be evaluated analytically as follow (in a similar way we can solve the second one):

### ( ) ( )

### ( ) ( )

### ( )

### 0

### , , , ,

### 2 2

### , , , , ,

### 2 2

### , , 2

*C*

*B*

*C*

*bx* *x*

*m* *m*

*m* *ax* *m*

*S* *cx d* *x*

*m* *m*

*m* *m*

*ax* *x* *S*

*m* *S* *m*

*l x* *l x*

*x y z* *dxdy* *x y z* *dy*

*x* *A* *A*

*l x* *l*

*x y z* *dy* *x y z* *dxdy*

*A* *A*

*x y z* *l* *dxdy* *A*

### φ φ

### φ φ

### φ

±

±

±

### ± ±

### − +

### ± ±

### ±

###

### ∂ =

### ∂

###

###

### + − =

###

###

### −

## ∫∫ ∫

## ∫ ∫∫

## ∫∫

### (C.4)

### since:

### ( )

### ( ) ( )

### , , 2

### .

### , , , , 0

### 2 2

*B*

*B*

*C* *C*

*C* *C*

*cx* *d*

*m* *B* *B*

*ax* *m*

*B* *B*

*bx* *cx* *d*

*m C* *m C*

*C* *C*

*m* *m*

*ax* *ax*

*C* *C*

*x* *y z* *l x* *dy* *A*

*ax* *cx* *d*

*l x* *l x*

*x* *y z* *dy* *x* *y z* *dy*

*A* *A*

*bx* *cx* *d*

### φ

### φ φ

### − +

### ±

### − +

### ± ±

### = − +

### − =

### = − +

## ∫

## ∫ ∫

### (C.5)

### For the relation (C.5) and (2.21), the (C.1) becomes:

### ( )

### ( ) ( )

### ( )

### , , ,

### , , , , .

### , , ,

*m* *m*

*S* *m*

*m* *m*

*m* *m*

*S* *S*

*s* *m* *s* *m*

*S*

*x y z* *l* *dS* *A*

*l* *l*

*x y z* *dxdy* *x y z* *dxdy*

*A* *A*

*x y z* *dS*

### φ ξ φ

### φ φ

### φ ξ φ ξ

±

+ −