Tensors for Physics

8.2 Surface Integrals, Stokes 127 8.3 Exercise: Verify the Stokes Law for a Vorticity Field Compute the integrals on both sides ...

128 8 Integration of Fields The line integral is over the circle with radiusR. For symmetry reasons, theH-field is tangential, i ...

8.2 Surface Integrals, Stokes 129 over the areaA, bounded by the wire and application of the Stokes law (8.36) yields the equati ...

130 8 Integration of Fields HereJis theJacobi determinant, also calledfunctional determinant, which can be computed according to ...

8.3 Volume Integrals, Gauss 131 As a more specific example, a half-sphere with radiusR, located above thex–y- plane, is chosen. ...

132 8 Integration of Fields For (8.54) one findsMV=mNV,for(8.55) the result is MV= ∑NV i= 1 mi, NVis the number of particles loc ...

8.3 Volume Integrals, Gauss 133 integrals to be evaluated are of the form ∫ V ...d^3 r= 2 π ∫a 2 a 1 r^2 dr ∫θmax 0 sinθdθ...= 2 ...

134 8 Integration of Fields 8.3.3 Application: Moment of Inertia Tensor The moment of inertia tensorΘμν, introduced in Sect.4.3. ...

8.3 Volume Integrals, Gauss 135 Fig. 8.15Brick stone with sidesa,b,c. The coordinate axes coincide with the principal axes of th ...

136 8 Integration of Fields For a sphere with radiusa/2, the tensor has the same form, just withΘ= 101 Ma^2. Obviously, on the l ...

8.3 Volume Integrals, Gauss 137 Fig. 8.16The potato of Prof. Muschik as integration volume,nis the outer normal vector, perpendi ...

138 8 Integration of Fields ∫ ̂rμ̂rνd^2 ̂r= 4 π 3 δμν is obtained. The expressions just computed for the volume and surface inte ...

8.3 Volume Integrals, Gauss 139 Therelation(8.77) is referred to as the differential form of theGauss law. The integral of (8.77 ...

140 8 Integration of Fields 8.3.6 Integration by Parts. Letf=f(r)andg=g(r)be two field functions. Due to the product rule∇μ(gf)= ...

8.4 Further Applications of Volume Integrals 141 d dt MV+ ∫ ∂V nνjνd^2 s= 0 , (8.86) wherenis the outer normal vector on the sur ...

142 8 Integration of Fields 8.4.2 Momentum Balance, Force on a Solid Body The local conservation equation for the linear momentu ...

8.4 Further Applications of Volume Integrals 143 Fig. 8.17The tangential force density due to non-diagonal elements of the press ...

144 8 Integration of Fields isρg, whereρis the mass density of the liquid, andgis the gravity acceleration, pointing downward, t ...

8.5 Further Applications in Electrodynamics 145 8.5 Further Applications in Electrodynamics 8.5.1 Energy and Energy Density in E ...

146 8 Integration of Fields cf. Sects.2.6.4and5.3.4. Then the energy density is equal to u= 1 2 EνDν= 1 2 ε 0 ενμEμEν. (8.104) C ...