Tensors for Physics

106 7 Fields, Spatial Differential Operators the equation (7.78) can be written as ∇μ=r̂μ ∂ ∂r −r−^1 εμνλr̂νLλ. (7.81) Notice: t ...

7.6 Rules for Nabla and Laplace Operators 107 Side Remark: Where Does the Expression for the Linear Momentum Operator Come From? ...

108 7 Fields, Spatial Differential Operators 7.6.3 Radial and Angular Parts of the Laplace Operator. The Laplace operator can be ...

7.6 Rules for Nabla and Laplace Operators 109 momentum corresponds to the operatorpop=i∇,cf.(7.85). Hence the Hamilton operator ...

Chapter 8 Integration of Fields Abstract The integration of fields is treated in this chapter. Firstly, line integrals are consi ...

112 8 Integration of Fields Here drμis the Cartesian component of the differential change dralong the curve C. The line integral ...

8.1 Line Integrals 113 Fig. 8.1 Two curvesC 1 and C 2 starting and ending at the same points ∫ C 1 f(r)drμ− ∫ C 2 f(r)drμ= ∮ C f ...

114 8 Integration of Fields Fig. 8.2 Line integral yields a relative position vector (ii) Vector Fields Now letfbe the component ...

8.1 Line Integrals 115 or I=Φ(r 2 )−Φ(r 1 ). (8.7) For the case of a vector field obtained as a gradient of a potential, the lin ...

116 8 Integration of Fields Remark: in physics, forcesFassociated with a potential are given by the negative gradient of the pot ...

8.2 Surface Integrals, Stokes 117 8.2 Surface Integrals, Stokes 8.2.1 Parameter Representation of Surfaces Surfaces in 3D space ...

118 8 Integration of Fields 8.2.2 Examples for Parameter Representations of Surfaces (i) Plane Leteandube two orthogonal unit ve ...

8.2 Surface Integrals, Stokes 119 Fig. 8.6 Cylinder coordinates with the vectors eφ,ezandeρ with the two parametersφandz. The ta ...

120 8 Integration of Fields Fig. 8.7 Spherical polar coordinates with the vectors eφ,eθander 8.2.3 Surface Integrals as Integral ...

8.2 Surface Integrals, Stokes 121 Fig. 8.8 Surface in real space and in thep–qparameter plane Often the symbol ∮ is used to indi ...

122 8 Integration of Fields Fig. 8.9 Planar polar coordinates for a segment of a planar ring For the simple casef =1 and an inte ...

8.2 Surface Integrals, Stokes 123 The surface integral over a regionAon the surface of the sphere is Sμ=R^2 ∫ A f(r(θ, φ))̂rμ(θ, ...

124 8 Integration of Fields Fig. 8.10Flux through an areaAand side view of an effective areaAeff S=vAeff. Herevis the magnitude ...

8.2 Surface Integrals, Stokes 125 It is understood that the closed curveC=∂Ais the rim, or the contour line, of the surfaceA. Th ...

126 8 Integration of Fields Insertion of this expression intoSμ= ∫ Aελνμ∇νf(r)dsλ, which is the left hand side of (8.35), leads ...