A Norms A.l Vector Norms Definition A.l. Vector norm: A vector norm on mn is a function II II : mn ---+ lR, which fulfills the following properties: (i) llxll 2: 0 for all x E lRn (ii) llxll = 0 iff x = 0 (iii) llx + Yll :::; llxll + IIYII (iv) llaxll = lalllxll We use the Holder or p-norms, which are defined by (A.l) Therefore, we compute, e.g., the 1-norm, 2-norm, and oo-norm as follows (A.2) i=l n ) 1/2 llxll2 = ( ~ lxil2 (A.3) llxlloo = maxilxil· (A.4) The p-norms have the following useful properties • lxTyl:::; llxiiPIIYIIq, ~ + ~ = 1 (Holder inequality) • lxTyl:::; llxlb IIYII2 (Cauchy-Schwarz inequality) • llxll2 :::; llxllt :::; vlnllxll2 270 A Norms • llxlloo :S llxll2 :S vfnllxlloo • llxlloo :S llxll1 :S nllxlloo With the help of norms we can define a distance on a vector space, and furthermore, we call a vector space with a norm a normed space. A.2 Matrix Norms Definition A.2. Matrix norm: A matrix norm on 1Rnxm is a function II II : 1Rnxm -+ 1R, which fulfills the following properties: (i) IIAII ~ 0 for all A E 1Rnxm (ii) IIAII = 0 iff A= 0 (iii) IlA + Bll :S IIAII + IIBII for all A, BE 1Rnxm (iv) llaAII = laiiiAII for all a E 1R and A E 1Rnxm The matrix norm associated to the vector p-norm is defined by the operator norm (A.5) Other matrix norms are Frobenius or F-norm (A.6) n column sum norm (A.7) i=l m IIAIIoo = maxi L laij I row sum norm . (A.8) j=l B Scalar and Vector Fields Definition B. I. (Scalar field) If we assign to each point in ffi3 defined by the vector r a scalar quantity V (r) (e.g., electric potential, temperature, acous­ tic velocity potential}, then V is called a scalar field. For the illustration of scalar fields we use equipotential lines in the 2D case and equipotential surfaces in the 3D case, where the scalar quantity V(r) is constant (Fig. B.l). v> Fig. B.l. Illustration of a scalar field V with the help of equipotential surfaces Definition B.2. (Vector field) If we assign to each point ffi3 defined by the vector r a vector quantity F(r) (e.g., electric field, magnetic field, mechanical deformation}, then F is called a vector field. 272 B Scalar and Vector Fields Vector fields are divided into irrotational vector fields (e.g., electrostatic field) and solenoidal vector fields (e.g., magnetic field) as shown in Fig. B.2 (see also Sects. B.12 and B.13). y y a) b) Fig. B.2. (a) Solenoidal vector field; (b) Irrotational vector field Fig. B.3. Lines of force for the vector field F(r) The lines of force (see Fig. B.3) are defined by F(r) x dr = 0, (B.l) which means that in each point of the lines the tangential vector is parallel to the field vector F. In the following, we try to compute the lines of force for the vector field r F(r) = 3 (B.2) r with the help of (B.l). Since we are just interested in the direction, we have to solve rxdr=O. (B.3) By using a Cartesian coordinate system, we obtain B Scalar and Vector Fields 273 r = xex + yey + zez (B.4) dr = dxex + dyey + dzez , (B.5) (B.6) and ex ey ez (ydz- zdy) rx dr= x y z = zdx-xdz (B.7) dx dy dz xdy- ydx Therefore, we can formulate the following three relations ydz = zdy (B.8) zdx = xdz (B.9) xdy = ydx. (B.lO) We now search for the line of force including point P0 (x0 , y0 , z0 ). Integration of (B.8) results in y z ln-=ln- (B.ll) Yo zo and zoy = YoZ. (B.l2) Analogously, we can compute the solutions of the other two differential equa­ tions ZoX = XoZ (B.13) YoX = xoy · (B.14) From (B.13) a plane through point Po containing they-axis, and from (B.14) a plane through point Po containing the z-axis is defined. The intersection of the two planes leads to a straight line through the origin, and therefore we obtain the vector field drawn in Fig. B.4, which corresponds, e.g., to the vector field of an electric charge. Fig. B.4. Lines of force of the vector field F(r) = r/r3 274 B Scalar and Vector Fields B.l The Nabla (V) Operator First, we recall that a scalar function may depend on one or more variables, e.g., using Cartesian coordinates, a function can be denoted by f = f(x,y,z). The partial derivatives read as of of of ox' 8y' 8z. The nabla operator V is defined in Cartesian coordinates by (B.15) where ex, ey and ez are the unit vectors in x-, y-, and z-directions. The interaction between the nabla operator and a scalar or a vector field yields its geometric significance. B.2 Definition of Gradient, Divergence, and Curl We introduce a scalar function V with nonzero first-order partial derivatives with respect to the coordinates x, y, and z, and a vector field F with compo­ nents Fx, Fy, and Fz. Then, the following operations are defined: 1. Gradient of a scalar: As can be seen, the result of this operation is a vector. 2. Divergence of a vector: div F = V . F = 8Fx + 8Fy + 8Fz . ax 8y 8z Therefore, the result of this operation is a scalar value. B.3 The Gradient 275 3. Curl of a vector: !!E.._8y 8Fy)8z _ ( 8Fm _!!E.. curl F = \7 X F = a;ax 8/oy a;az - 8z 8x 8Fy _!!E.. Fx Fy Fz 8x 8y The result of taking the curl of a vector is again a vector. B.3 The Gradient We will consider the scalar function V(x,y,z) with its partial derivatives 8Vf8x, 8Vj8y, 8Vf8z and dependent on a point P = (x, y, z). In a first step we calculate the total differential of V av av av dV =ox dx+ oy dy+ oz dz. (B.16) Now, we define a point P' infinitely close toP by P' = (x+ dx, y+ dy, z+dz). By calculating the vector dP = P' - P, which has the components dP ( dx, dy, dz)T, we can write (B.l6) as (B.17) (B.l8) For the geometrical illustration of the gradient, consider an equipotential sur­ face, i.e., a surface with V = canst. (see Fig. B.5). Hence, for all differential vv dP p Fig. B.5. The gradient is orthogonal to a constant potential surface displacements from P to P' on this surface dV = 0 holds, and therefore, VV · dP = 0. (B.19) 276 B Scalar and Vector Fields From the definition of the scalar product it is clear that VV and dP are orthogonal. In this situation the displacement from P to P' points into the direction of increasing V, as shown in Fig. B.6, and the scalar product VV · dP is positive. vV p Fig. B.6. Geometrical representation of the gradient From the foregoing arguments, we conclude that VV is a vector, perpen­ dicular to the surface on which V is constant and that it points in the direction of increasing V. As an example we consider a function r(x, y, z), which defines the distance of a point P from the origin (0, 0, 0). The surface r = con st. is a sphere of radius r with center (0, 0, 0), whose equation is given by r = J x2 + y2 + z2 . Therefore, the gradient calculates as 8r X X ax Jx2 + y2 + z2 r 8r y 8y r 8r z 8z r Vr = xex + yey + zez r r r Geometrically speaking, V r points in the direction of increasing r, or towards spheres with radii larger than r. B.4 The Flux 277 B.4 The Flux Definition B.3. (Flux) The vector field F(r) and a corresponding surfacer as shown in Fig. B. 7 are given. The vector n denotes the normal unit vector of the differential surface di'. Therefore, the differential flux d'ljJ through di' is defined by d'I/J=F·dr=F·ndr. (B.20) The total flux 'ljJ computes as '1/J= IF· dr. (B.21) r In the following, we want to compute the flux 'ljJ of the vector field F(r) = r Fig. B.7. Flux through the surfacer through the square r with side length h according to Fig. B.8. With the normal unit vector n = ex and dr = dy dzex we obtain h h 'ljJ = I I (-hex + yey + zez) ·ex dy dz 0 0 h h = -h 11 dydz 0 0 = -h3. (B.22) The total flux 'ljJ through a closed surface S is given by 278 B Scalar and Vector Fields "' z y t Jf h h -h Fig. B.S. Flux 'lj; through the square with area h2 (B.23) and defines whether we have sources ('1/J > 0) or sinks ('1/J < 0) within r. A very important property of the flux 'ljJ defined by a closed surface is given by (see Fig. B.9) f F-dr+ f F-dr= f F-dr. (B.24) r1uro r2uro nur2 Fig. B.9. Flux through the closed surface HUH B.5 Divergence Definition B.4. (Divergence) The vector field F(r) is given. If we divide the flux 'ljJ, defined by a closed surface r, by the corresponding volume Jl and let the volume Jl tend to zero, then the obtained value is called the divergence (source density) B.5 Divergence 279 divF = N!!:o ~iF· dr = 1~ lr.
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