Exempelsamling Vektoranalys

58h) alt. II:tyochh) alt. III:rot((a × r) × b) = {(8.24)} = (b · ∇)(a × r) − b(∇ · (a × r)) == {ex. 64 f) och (8.23)} == a × (b · ∇)r + b(a · (∇ × r)) = a × b()∂(b · ∇)r = b x∂x + b ∂y∂y + b ∂z (x, y, z) = (b x , b y , b z ) = b∂z∇ × r =∣e x e y e z∂ ∂ ∂∂x ∂y ∂zx y z= 0∣rot((a × r) × b) = rot(b × (r × a)) = rot((b · a)r − (b · r)a) == {(8.22)} = (b · a)(∇ × r) −(∇(b · r)) × a =} {{ }=0= −b × atyeller∇(b · r) =( ∂∂x , ∂ ∂y , ∂ )(b x x + b y y + b z z) = (b x , b y , b z ) = b∂z∇(b · r) = {(8.25)} = (b · ∇)r +b × (∇ × r)} {{ } } {{ }=b =066. 66a)[grad(a · gradφ)] i = ∂ i (a j ∂ j φ) = a j ∂ i ∂ j φ = [(a · ∇)∇φ] ib)[rot(a × gradφ)] i = ǫ ijk ∂ j (a × ∇φ) k =Alltså: A = −B = (a · ∇)∇φc) Medblir= ǫ ijk ǫ klm ∂ j (a l ∂ m φ) == (δ il δ jm − δ im δ jl )a l ∂ j ∂ m φ == a i ∂ m ∂ m φ − a l ∂ l ∂ i φ = [a ∇ 2 φ −(a · ∇)∇φ]}{{}i=0∇φ = e x yz + e y xz + e z xyA = e x (a y z + a z y) + e y (a x z + a z x) + e z (a x y + a y x)