∇→×F→=(1r∂Fz∂φ−∂Fφ∂z)e→r+(∂Fr∂z−∂Fz∂r)e→φ+(Fφr+∂Fφ∂r−1r∂Fr∂φ)e→z

si300_e  (Eq. 7.97)

∇→×F→=(1r∂Fz∂φ−∂Fφ∂z)e→r+(∂Fr∂z−∂Fz∂r)e→φ+(∂∂r(rFφ)−1r∂Fz∂φ+Fφr)e→z

si301_e  (Eq. 7.98)

Eq. 7.97 gives the curl in cylindrical coordinates. Again, Eq. 7.98 is a slightly more compact notation of the same equation.

7.6.1.4 Scalar Laplacian

The next conversion we require is the Laplace operator in cylindrical coordinates. Here, we can exploit a small trick due to the fact that

Δψ=∇→⋅(∇→ψ)

si302_e

where we can use the gradient of ψ given by Eq. 7.90 as

∇→ψ=∂ψ∂re→r+1r∂ψ∂φe→φ+∂ψ∂ze→z

si303_e

and the divergence by Eq. 7.93 as

Δψ=∇→⋅(∇→ψ)=1r∂ψ∂r+∂∂r(∂ψ∂r)+1r∂∂φ(1r,∂ψ∂φ)+∂∂z(∂ψ∂z)=∂2ψ∂r2+12∂ψ∂r+1r2∂2ψ∂φ2+∂2ψ∂z2

si304_e  (Eq. 7.99)

=1r∂∂r(r∂ψ∂r)+1r2∂2ψ∂φ2+∂2ψ∂z2

si305_e  (Eq. 7.100)

Eq. 7.99 is the Laplace operator in cylindrical coordinates. Eq. 7.100 is a commonly encountered, slightly more compact notation

7.6.1.5 Vector Laplacian

The second commonly encountered version of the Laplacian is the vector Laplacian of the vector F→si1_e. For transferring into cylindrical coordinates we will make use of Eq. 7.6. For this equation we require the two terms ∇→(∇→F→)si307_e and ∇→×(∇→×F→)si308_e. The first can be derived by first calculating the divergence ∇→F→si309_e given by Eq. 7.93 as

∇→F→=Frr+∂Fr∂r+1r∂Fφ∂φ+∂Fz∂z

si310_e

that we then use to form the gradient ∇→(∇→F→)si307_e given by Eq. 7.90 as

∇→(∇→F→)=(∂∂r(Frr+∂Fr∂r+1r∂Fφ∂φ+∂Fz∂z))e→r+(1r∂∂φ(Frr+∂Fr∂r+1r∂Fφ∂φ+∂Fz∂z))e→φ+(∂∂z(Frr+∂Fr∂r+1r∂Fφ∂φ+∂Fz∂z))e→z=(−1r2Fr+1r∂Fr∂r+∂Fr∂r2−1r2∂2Fφ∂φ+1r∂2Fφ∂φ∂r+∂2Fz∂z∂r)e→r+(1r(1r∂Fr∂φ+∂2Fr∂r∂φ+1r∂2Fφ∂φ2+∂2Fz∂z∂φ))e→φ+(1r∂Fr∂z+∂2Fr∂r∂z+1r∂2Fφ∂φ∂z+∂2Fz∂z)e→z

si312_e  (Eq. 7.101)

The second term we require for Eq. 7.6 is ∇→×(∇→×F→)si308_e that can be obtained by applying Eq. 7.97 twice to the vector F→si1_e finding

∇→×(∇→×F→)=∇→×((1r∂Fz∂φ−∂Fφ∂z)e→r+(∂Fr∂r−∂Fz∂r)e→φ+(Fφr+∂Fφ∂r1r∂Fr∂φ)e→z)=(1r∂∂φ(Fφr+∂Fφ∂r−1r∂Fr∂φ)−∂∂z(∂Fr∂z−∂Fz∂r))e→r+(∂∂z(1r∂Fz∂φ−∂Fφ∂z)−∂∂r(Fφr+∂Fφ∂r−1r∂Fr∂φ))e→φ+(1r(∂Fr∂z−∂Fz∂r)+∂∂r(∂Fr∂z−∂Fz∂r)−1r∂∂φ(1r∂Fz∂φ−∂Fφ∂z))e→z=(1r(1r∂Fφ∂φ+∂2Fφ∂r∂φ−1r∂2Fr∂φ2)−∂2Fr∂φ2−∂2Fz∂r∂z)e→r+(1r∂2Fz∂φ∂z−∂2Fφ∂z2+1r2Fφ−1r∂Fφ∂r−∂2Fφ∂r2−1r2∂Fr∂φ+1r∂2Fr∂φ∂r)e→φ=(1r(∂Fr∂z−∂Fz∂r)+∂2Fr∂z∂r−∂2Fz∂r2−1r(1r∂2Fz∂φ2−∂2Fφ∂z∂φ))e→z

si315_e  (Eq. 7.102)

Using Eq. 7.101 and Eq. 7.102 we can now reassemble Eq. 7.6 to

ΔF→ΔF→=∇→(∇→F→)−∇→×(∇→×F→)=(−1r2Fr+1r∂Fφ∂r+∂2Fφ∂r2−1r2∂Fφ∂φ+1r∂2Fφ∂φ∂r+∂2Fz∂z∂r−(1r(1r∂Fφ∂φ+∂2Fφ∂r∂φ−1r∂2Fr∂φ2)−∂2Fr∂z2+∂2Fφ∂r∂φ))e→r+(1r(1r∂Fr∂φ+∂2Fφ∂r∂φ+1r∂2Fφ∂φ2+∂2Fφ∂r∂φ)−(1r∂2Fφ∂φ∂z−∂2Fφ∂z2+1r2Fφ−1r∂Fφ∂r−∂2Fφ∂r2−1r2∂Fr∂φ+1r∂2Fr∂φ∂r))e→φ+(1r∂Fr∂z+∂2Fr∂r∂z+1r∂2Fφ∂φ∂z+∂2Fz∂z2−(1r(∂Fr∂z−∂Fz∂r)+∂2Fr∂z∂r−∂2Fz∂r2−1r(1r∂2Fz∂φ2−∂2Fφ∂z∂φ)))e→z=(∂2Fr∂r2+1r∂Fr∂r+1r2∂2Fr∂φ2+∂2Fr∂z2−Frr2−2r2∂Fφ∂φ)e→r++(∂2Fφ∂r2+1r∂Fφ∂r+1r2∂2Fφ∂φ2+∂2Fφ∂z2−Fφr2+2r2∂Fr∂φ2)e→φ+(∂2Fz∂z2+1r∂Fz∂r+1r2∂2Fz∂φ2+∂2Fz∂r2)e→z

si316_e  (Eq. 7.103)

Eq. 7.103 is the vector Laplacian in cylindrical coordinates. If you look very carefully, you will see that the first four terms of each of the vector components are nothing other than the scalar Laplacians ΔFr, ΔFφ and ΔFz which allows a slightly more compact notation in the form of

ΔF→=(ΔFr−Frr2−2r2∂Fφ∂φ)e→r+(ΔFφ−Fφr2+2r2∂Fr∂φ)e→φ+(ΔFz)e→z

si317_e  (Eq. 7.104)

7.6.1.6 Material Derivative

The last operator we will convert is the material derivative (see section 7.1.3.10) which we will require, e.g., for the Navier–Stokes equation (see Eq. 11.40). For this we will first require the scalar product F→⋅∇→si318_e of the vector

F→=⎛⎝⎜FrFφFz⎞⎠⎟=⎛⎝⎜cosφFr−sinφFφsinφFr+cosφFφFz⎞⎠⎟

si319_e

that we converted to Cartesian coordinates according to Eq. 7.32. We now carry out the scalar product in Cartesian coordinates finding

F→⋅∇→=⎛⎝⎜cosφFr−sinφFφsinφFr+cosφFφFz⎞⎠⎟⋅⎛⎝⎜⎜∂∂x∂∂y∂∂z⎞⎠⎟⎟

si320_e  (Eq. 7.105)

where we now require the partial differentials ∂∂x,∂∂y,and∂∂zsi321_e that we can simply copy from Eq. 7.86, Eq. 7.87, and Eq. 7.88 by simply omitting ψ. Using these equations, Eq. 7.86 becomes

F→⋅∇→F→⋅∇→====⎛⎝⎜cosφFr−sinφFφsinφFr+cosφFφFz⎞⎠⎟⋅⎛⎝⎜⎜cosφ∂∂r−sinφr∂∂φsinφ∂∂r+cosφr∂∂φ∂∂z⎞⎠⎟⎟(cosφFr−sinφFφ)(cosφ∂∂r−sinφr∂∂φ)+(sinφFr+cosφFφ)(sinφ∂∂r+cosφr∂∂φ)+Fz∂∂zFr(cosφ(cosφ∂∂r−sinφr∂∂φ)+sinφ(sinφ∂∂r+cosφr∂∂φ))+Fφ(−sinφ(cosφ∂∂r−sinφr∂∂φ)+cosφ(sinφ∂∂r+cosφr∂∂φ))+Fz∂∂zFr∂∂r+Fφr∂∂φ+Fz∂∂z

si322_e

Please note that the result is a scalar. It is the intuitive result of the scalar multiplication of F→r,φ,z⋅∇→r,φ,zsi323_e. In the next step we multiply this expression with the vector G→si56_e that we again convert to Cartesian coordinates finding

(F→⋅∇→)G→====(Fr∂∂r+Fφr∂∂φ+Fz∂∂z)⎛⎝⎜cosφGr−sinφGφsinφGr+cosφGφGz⎞⎠⎟((Fr∂∂r+Fφr∂∂φ+Fz∂∂z)(cosφGr−sinφGφ))e→x+((Fr∂∂r+Fφr∂∂φ+Fz∂∂z)(sinφGr+cosφGφ))e→y+(Fr∂∂r+Fφr∂∂φ+Fz∂∂z)Gze→z(Fr∂∂r(cosφGr−sinφGφ)+Fφr∂∂φ(cosφGr−sinφGφ)+Fz∂∂z(cosφGr−sinφGφ))e→x+(Fr∂∂r(sinφGr+cosφGφ)+Fφr∂∂φ(sinφGr+cosφGφ)+Fz∂∂z(sinφGr+cosφGφ))e→y+(Fr∂Gz∂r+Fφr∂Gz∂φ+Fz∂Gz∂z)e→z(Fr(cosφ∂Gz∂r−sinφ∂Gφ∂r)+Fφr(−sinφGr+cosφ∂Gr∂φ)−cosφGφ−sinφ∂Gφ∂φ)+Fz(cosφ∂Gr∂z−sinφ∂Gφ∂z))e→x+(Fr(sinφ∂Gr∂r+cosφ∂Gφ∂r)+Fφr(cosφGr+sinφ∂Gr∂φ−sinφGφ+cosφ∂Gφ∂φ)+Fz(sinφ∂Gr∂r+cosφ∂Gφ∂z))e→y+(Fr∂Gz∂r+Fφr∂Gz∂φ+Fz∂Gz∂z)e→z

si325_e

where we reconvert the result into cylindrical coordinates using Eq. 7.28, Eq. 7.29, and Eq. 7.30 to find

(F→⋅∇→)G→=(Fr(cosφ∂Gr∂r−sin∂Gφ∂r))+Fφr(sinφGr+cos∂Gr∂φ−cosφGφ−sinφ∂Gφ∂φ)+Fz(cosφ∂Gr∂z−sinφ∂Gφ∂z))(cosφe→r−sinφe→φ)−sinφGφ+cosφ∂Gφ∂φ(Fr(sinφ∂Gr∂r+cosφ∂Gφ∂r)+Fφr(cosφGr+sinφ∂Gr∂φ−sinφGφ+cosφ∂Gφ∂φ+Fz(sinφ∂Gr∂z−cosφ∂Gφ∂z))(sinφe→r−cosφe→φ)+(Fz∂Gr∂r+Fφr+Fz∂Gr∂z)e→z=(Fr(cosφ∂Gr∂r−sinφ∂Gφ∂r)+Fφr(−sinφcosφGr+cos2φ∂Gr∂φ−cos2φGφ−sinφcosφ∂Gφ∂φ)+Fz(cos2φ∂Gr∂z−sinφcosφ∂Gφ∂z)+Fr(sin2φ∂Gr∂r+sinφcosφ∂Gφ∂r)+Fφr(sinφcosφGr+sin2φ∂Gφ∂r−sin2φGφ+sinφcosφ∂Gφ∂r)+Fz(sin2φ∂Gr∂z+sinφcosφ∂Gφ∂z)e→r+(Fr(−sinφcosφ∂Gr∂r+sin2φ∂Gφ∂r)+Fφr(sin2φGr−sinφcosφ∂Gr∂φ+sinφcosφ+sin2∂Gφ∂φ)+Fz(−sinφcosφ∂Gr∂z+sin2φ∂Gφ∂z)+Fr(sinφcosφ∂Gr∂r+cos2φ∂Gφ∂r)+

si326_e

(F→⋅∇→)G→Fφr(cos2φGr+sinφcosφ∂Gr∂φ−sinφcosφGφ+cos2φ∂Gφ∂φ)+Fz(sinφcosφ∂Gr∂z+cos2φ∂Gφ∂z)e→φ+(Fr∂Gz∂r+Fφr∂Gz∂φ+Fz∂Gz∂z)e→z=(Fr∂Gr∂r+Fφr∂Gr∂φ+Fz∂Gr∂z−FφGφr)e→r+(Fr∂Gφ∂r+Fφr∂Gφ∂φ+Fz∂Gφ∂z−FφGrr)e→φ+(Fr∂Gz∂r+Fφr∂Gz∂φ+Fz∂Gz∂z)e→z

si505_e  (Eq. 7.106)

Eq. 7.106 is the sought material derivative in cylindrical coordinates.

7.6.2 Polar Coordinates

We can skip the derivations in polar coordinates again, due to the fact that we can simply insert the operators in cylindrical coordinates while ignoring the contribution along the z-axis.

7.6.2.1 Gradient

The gradient in polar coordinates according to the simplified version of Eq. 7.90 is given by

Δ→ψ=∂ψ∂re→r+1r∂ψ∂φe→φ

si327_e  (Eq. 7.107)

7.6.2.2 Divergence

Likewise, the divergence can be derived from Eq. 7.93 as

∇→⋅F→=Frr+∂F∂r+1r∂Fφ∂φ

si328_e  (Eq. 7.108)

7.6.2.3 Curl

In analogy, the curl is given by simplifying Eq. 7.97 as

∇→×F→=(1r∂Fz∂φ−∂Fφ∂z)e→r+(∂Fr∂z−∂Fz∂r)e→φ

si329_e  (Eq. 7.109)

7.6.2.4 Scalar Laplacian

The scalar Laplacian is derived from Eq. 7.99 as

Δψ=∂2ψ∂r2+1r∂ψ∂r+1r2∂2ψ∂φ2

si330_e  (Eq. 7.110)

7.6.2.5 Vector Laplacian

Likewise, the vector Laplacian can be derived from Eq. 7.104 as

ΔF→=(ΔFr−Frr2−2r2∂Fφ∂φ)e→r+(ΔFφ−Fφr2+2r2∂Fr∂φ)e→φ

si331_e  (Eq. 7.111)

7.6.2.6 Material Derivative

Finally, the material derivative is given from simplifying Eq. 7.106 to

(F→⋅∇→)G→=(Fr∂Gr∂r+Fφr∂Gr∂φ−FφGφr)e→r+(Fr∂Gφ∂r+Fφr∂Gφ∂φ+FφGrr)e→φ

si332_e  (Eq. 7.112)

7.6.3 Spherical Coordinates

7.6.3.1 Gradient

Finally, we will derive the operators in spherical coordinates (see Fig. 7.4d). Again, we will begin with the gradient of the scalar ψ defined as

∇→ψ=∂ψ∂xe→x+∂ψ∂ye→y+∂ψ∂ze→z

si273_e  (Eq. 7.113)

where we again require the total differentials in order to express ψ(x, y, z) which is a function of x, y, and z in Cartesian coordinates as a function of ρ, θ, and φ in spherical coordinates. We find

∂ψ∂x=∂ψ∂ρ∂ρ∂x+∂ψ∂θ∂θ∂x+∂ψ∂φ∂φ∂x

si334_e  (Eq. 7.114)

∂ψ∂y=∂ψ∂ρ∂ρ∂y+∂ψ∂θ∂θ∂y+∂ψ∂φ∂φ∂y

si335_e  (Eq. 7.115)

∂ψ∂z=∂ψ∂ρ∂ρ∂z+∂ψ∂θ∂θ∂z+∂ψ∂φ∂φ∂z

si336_e  (Eq. 7.116)

where we take the partial derivatives from the inverse Jacobian (see Eq. 7.72) from which we find

∂ρ∂x=sinθcosφ

si337_e

∂ρ∂y=sinθsinφ

si338_e

∂ρ∂z=cosθ

si339_e

∂θ∂x=cosθcosφρ

si340_e

∂θ∂y=cosθsinφρ

si341_e

∂θ∂z=−sinθρ

si342_e

∂φ∂x=−1ρsinφsinθ

si343_e

∂φ∂y=1ρcosφsinθ

si344_e

∂φ∂z=0

si345_e

Reinserting these into Eq. 7.113 yields

∇→ψ=(∂ψ∂ρ∂ρ∂x+∂ψ∂θ∂θ∂x+∂ψ∂φ∂φ∂x)e→x+(∂ψ∂ρ∂ρ∂y+∂ψ∂θ∂θ∂y+∂ψ∂φ∂φ∂y)e→y+(∂ψ∂ρ∂ρ∂z+∂ψ∂θ∂θ∂z+∂ψ∂φ∂φ∂z)e→z=(∂ψ∂ρsinθcosφ+∂ψ∂θcosθsinφρ−∂ψ∂φ1ρsinφsinθ)e→x+(∂ψ∂ρsinθsinφ+∂ψ∂θcosθsinφρ+∂ψ∂φ1ρcosφsinθ)e→y+(∂ψ∂ρcosθ−∂ψ∂θsinθρ)e→z

si346_e  (Eq. 7.117)

where we now convert the basis vectors e→xsi138_e, e→ysi139_e, and e→zsi118_e of the Cartesian coordinate system to functions of the basis vectors e→rsi116_e, e→θsi123_e, and e→φsi117_e, of the cylindrical coordinate system using Eq. 7.50, Eq. 7.51, and Eq. 7.52 finding

∇→ψ=(∂ψ∂ρ∂ρ∂x+∂ψ∂θ∂θ∂x+∂ψ∂φ∂φ∂x)e→x+(∂ψ∂ρ∂ρ∂y+∂ψ∂θ∂θ∂y+∂ψ∂φ∂φ∂y)e→y+(∂ψ∂ρ∂ρ∂z+∂ψ∂θ∂θ∂z+∂ψ∂φ∂φ∂z)e→z=(∂ψ∂ρsinθcosφ+∂ψ∂θcosθcosφρ−∂ψ∂φ1ρsinφsinθ)(sinθcosφe→ρ+cosθcosφe→θ−sinφe→φ)+(∂ψ∂ρsinθsinφ+∂ψ∂θcosθsinφρ+∂ψ∂φ1ρcosφsinθ)(sinθsinφe→ρ+sinφcosθe→θ−cosφe→φ)+(∂ψ∂ρcosθ−∂ψ∂θsinθρ)(cosθe→ρ−sinθe→θ)=(sinθcosφ(∂ψ∂ρsinθcosφ+∂ψ∂θcosθcosφρ−∂ψ∂φ1ρsinφsinθ)+sinθsinφ(∂ψ∂ρsinθ−sinφ+∂ψ∂θcosθsinφρ+∂ψ∂φ1ρcosφsinθ)+cosθ(∂ψ∂ρcosθ−∂ψ∂θsinθρ))e→ρ+(cosθcosφ(∂ψ∂ρsinθcosφ+∂ψ∂θcosθcosφρ−∂ψ∂φ1ρsinφsinθ)+sinφcosθ(∂ψ∂ρsinθsinφ+∂ψ∂θcosθsinφρ+∂ψ∂φ1ρcosφsinθ)−sinθ(∂ψ∂ρcosθ−∂ψ∂θsinθρ))e→θ+(−sinφ(∂ψ∂ρsinθcosφ+∂ψ∂θcosθcosφρ−∂ψ∂φ1ρsinφsinθ)+cosφ(∂ψ∂ρsinθsinφ+∂ψ∂θcosθsinφρ+∂ψ∂φ1ρcosφsinθ))e→φ=(∂ψ∂ρsin2θcos2φ+∂ψ∂θsinθcosθcos2φρ−∂ψ∂φsinφcosφρ)+∂ψ∂ρsin2θsin2φ+∂ψ∂θsinθcosθsin2φρ+∂ψ∂φsinφcosφρ+∂ψ∂ρcos2θ−∂ψ∂θsinθcosθρ)e→ρ+(∂ψ∂ρsinθcosθcos2φ+∂ψ∂θcos2θcos2φρ−∂ψ∂φcosθsinφcosφρsinθ+∂ψ∂ρsinθcosθsin2φ+∂ψ∂θcos2θsin2φρ+∂ψ∂φcosθsinφcosφρsinθ−∂ψ∂ρsinθcosθ+∂ψ∂θsin2φρ)e→θ+(−∂ψ∂ρsinθsinφcosφ−∂ψ∂θcosθsinφcosφρ+∂ψ∂φsin2φρsinθ+∂ψ∂ρsinθsinφcosφ+∂ψ∂θcosθsinφcosφρ+∂ψ∂φcos2φρsinθ)e→φ=(∂ψ∂ρ(sin2θcos2φ+sin2θsin2φ+cos2θ)+∂ψ∂θ(sinθcosφcos2φρ+sinθcosθsin2φρ−sinθcosθρ)+∂ψ∂θ(−sinφcosφρ+sinφcosφρ))e→ρ

si353_e

∇→ψ+(∂ψ∂ρ(sinθcosθcos2φ+sinθcosθsin2φ−sinθcosθ)+∂ψ∂θ(cos2θcos2φρ+cos2θsin2φρ+sin2θρ)+∂ψ∂θ(−cosθsinφcosφρsinθ+cosθsinφcosφρsinθ))e→θ+(∂ψ∂ρ(−sinθsinφcosφ+sinθsinφcosφ)+∂ψ∂θ(−cosθsinφcosφρ+cosθsinφcosφρ)+∂ψ∂φ(sin2φρsinθ+cos2φρsinθ))e→φ=∂ψ∂ρe→ρ+1ρ∂ψ∂θe→θ+1ρsinθ∂ψ∂φe→φ

si504_e  (Eq. 7.118)

Eq. 7.118 gives the gradient in spherical coordinates.

7.6.3.2 Divergence

Next we derive the divergence ∇→⋅F→si2_e of the vector

F→=Fρe→ρ+Fθe→θ+Fφe→φ=(Fρsinθcosφ+Fθcosθcosφ−Fφsinφ)e→x+(Fρsinθsinφ+Fθcosθsinφ−Fφcosφ)e→y+(Fρcosθ−Fθsinθ)e→z

si355_e  (Eq. 7.119)

in spherical coordinates where we have used Eq. 7.53 to convert the basis vectors. According to Eq. 7.3 the divergence is given by

∇→⋅F→=∂∂x(Fρsinθcosφ+Fθcosθcosφ−Fφsinφ)+∂∂y(Fρsinθsinφ+Fθcosθsinφ−Fφcosφ)+∂∂z(Fρcosθ−Fθsinθ)=∂Fρ∂xsinθcosφ+Fρ(∂sinθ∂xcosφ−sinθ∂cosφ∂x)+∂Fθ∂xcosθcosφ+Fθ(∂cosθ∂xcosφ+cosθ∂cosφ∂x)−(∂Fφ∂xsinφ+Fφ∂sinφ∂x)+∂Fρ∂ysinθsinφ+Fρ(∂sinθ∂ysinφ+sinθ∂sinφ∂y)+∂Fθ∂ycosθsinφ+Fθ(∂cosθ∂ysinφ+cosθ∂sinφ∂y)+∂Fφ∂ycosφ+Fφ∂cosφ∂y+∂Fρ∂zcosθ+Fρ∂cosθ∂z−∂Fθ∂zsinθ−Fθ∂sinθ∂z

si356_e  (Eq. 7.120)

where we now again need to convert the partial differentials with respect to x, y, and z to partial differentials of ρ, θ, and φ for which we find

∂Fρ∂x=∂Fρ∂ρ∂ρ∂x+∂Fρ∂θ∂θ∂x+∂Fρ∂φ∂φ∂x=∂Fρ∂ρsinθcosφ+∂Fρ∂θ1ρcosθcosφ−∂Fρ∂θ1ρsinφsinθ

si357_e

∂Fρ∂y∂Fρ∂z∂Fθ∂x∂Fθ∂y∂Fθ∂z∂Fφ∂x∂Fφ∂y∂Fφ∂z∂sinθ∂x∂sinθ∂y∂sinθ∂z∂cosθ∂x∂cosθ∂y=∂Fρ∂ρ∂ρ∂y+∂Fρ∂θ∂θ∂y+∂Fρ∂φ∂φ∂y=∂Fρ∂ρsinθsinφ+∂Fρ∂θ1ρcosθsinφ+∂Fρ∂φ1ρcosφsinθ=∂Fρ∂ρ∂ρ∂z+∂Fρ∂θ∂θ∂z+∂Fρ∂φ∂φ∂z=∂Fρ∂ρcosθ−∂Fρ∂θ1ρsinθ=∂Fθ∂ρ∂ρ∂x+∂Fθ∂θ∂θ∂x+∂Fθ∂φ∂φ∂x=∂Fθ∂ρsinθcosφ+∂Fθ∂θ1ρcosθcosφ−∂Fθ∂φ1ρsinφsinθ=∂Fθ∂ρ∂ρ∂y+∂Fθ∂θ∂θ∂y+∂Fθ∂φ∂φ∂y=∂Fθ∂ρsinθsinφ+∂Fθ∂θ1ρcosθsinφ+∂Fθ∂φ1ρcosφsinθ=∂Fθ∂ρ∂ρ∂z+∂Fθ∂θ∂θ∂z+∂Fθ∂φ∂φ∂z=∂Fθ∂ρcosθ−∂Fθ∂θ1ρsinθ=∂Fφ∂ρ∂ρ∂x+∂Fφ∂θ∂θ∂x+∂Fφ∂φ∂φ∂x=∂Fφ∂ρsinθcosφ+∂Fφ∂θ1ρcosθcosφ−∂Fφ∂φ1ρsinφsinθ=∂Fφ∂ρ∂ρ∂y+∂Fφ∂θ∂θ∂y+∂Fφ∂φ∂φ∂y=∂Fφ∂ρsinθsinφ+∂Fφ∂θ1ρcosθsinφ+∂Fφ∂φ1ρcosφsinθ=∂Fφ∂ρ∂ρ∂z+∂Fφ∂θ∂θ∂z+∂Fφ∂φ∂φ∂z=∂Fφ∂ρcosθ−∂Fφ∂θ1ρsinθ=∂sinθ∂ρ∂ρ∂x+∂sinθ∂θ∂θ∂x+∂sinθ∂φ∂φ∂x=cos2θcosρρ=∂sinθ∂ρ∂ρ∂y+∂sinθ∂θ∂θ∂y+∂sinθ∂φ∂φ∂y=cos2θsinφρ=∂sinθ∂ρ∂ρ∂z+∂sinθ∂θ∂θ∂z+∂sinθ∂φ∂φ∂z=−sinθcosθρ=∂cosθ∂ρ∂ρ∂x+∂cosθ∂θ∂θ∂x+∂cosθ∂φ∂φ∂x=−sinθcosθcosφρ=∂cosθ∂ρ∂ρ∂y+∂cosθ∂θ∂θ∂y+∂cosθ∂φ∂φ∂y=−sinθcosθsinφρ

si657_e

∂cosθ∂z∂sinφ∂x∂sinφ∂y∂sin∂∂z∂cosφ∂x∂cosφ∂y∂cosφ∂z=∂cosθ∂ρ∂ρ∂z+∂cosθ∂θ∂θ∂z+∂cosθ∂φ∂φ∂z=sin2θρ=∂sinφ∂ρ∂ρ∂x+∂sinφ∂θ∂θ∂x+∂sinφ∂φ∂φ∂x=−1ρcosφsinφsinθ=∂sinφ∂ρ∂ρ∂y+∂sinφ∂θ∂θ∂y+∂sinφ∂φ∂φ∂y=1ρcos2φsinθ=∂sinφ∂ρ∂ρ∂z+∂sinφ∂θ∂θ∂z+∂sinφ∂φ∂φ∂z=0=∂cosφ∂ρ∂ρ∂x+∂cosφ∂θ∂θ∂x+∂cosφ∂φ∂φ∂x=1ρsin2φsinθ=∂cosφ∂ρ∂ρ∂y+∂cosφ∂θ∂θ∂y+∂cosφ∂φ∂φ∂y=−1ρsinφcosφsinθ=∂cosφ∂ρ∂ρ∂z+∂cosφ∂θ∂θ∂z+∂cosφ∂φ∂φ∂z=0

si358_e

Inserting the partial differentials into Eq. 7.120 yields

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