Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUSa is a vector field, these four combinations are grad(grad φ), div(div a), curl(div a)and grad(curl a). In each case the second (outer) vector operator is acting on thewrong type of field, i.e. scalar instead of vector or vice versa. In grad(grad φ), forexample, grad acts on grad φ, which is a vector field, but we know that grad onlyacts on scalar fields (although in fact we will see in chapter 26 that we can formthe outer product of the del operator with a vector to give a tensor, but that neednot concern us here).Of the five valid combinations of grad, div and curl, two are identically zero,namelycurl grad φ = ∇×∇φ = 0, (10.37)div curl a = ∇ · (∇×a) =0. (10.38)From (10.37), we see that if a is derived from the gradient of some scalar functionsuch that a = ∇φ then it is necessarily irrotational (∇ ×a = 0). We also notethat if a is an irrotational vector field then another irrotational vector field isa + ∇φ + c, whereφ is any scalar field and c is a constant vector. This followssince∇×(a + ∇φ + c) =∇×a + ∇×∇φ = 0.Similarly, from (10.38) we may infer that if b is the curl of some vector field asuch that b = ∇×a then b is solenoidal (∇ · b = 0). Obviously, if b is solenoidaland c is any constant vector then b + c is also solenoidal.The three remaining combinations of grad, div and curl arediv grad φ = ∇ · ∇φ = ∇ 2 φ = ∂2 φ∂x 2 + ∂2 φ∂y 2 + ∂2 φ∂z 2 , (10.39)grad div a = ∇(∇ · a),( ∂ 2 a x=∂x 2++ ∂2 a y∂x∂y + ∂2 a z∂x∂z) ( ∂ 2 )a xi +∂y∂x + ∂2 a y∂y 2 + ∂2 a zj∂y∂z( ∂ 2 )a x∂z∂x + ∂2 a y∂z∂y + ∂2 a z∂z 2 k, (10.40)curl curl a = ∇×(∇×a) =∇(∇ · a) −∇ 2 a, (10.41)where (10.39) and (10.40) are expressed in Cartesian coordinates. In (10.41), theterm ∇ 2 a has the linear differential operator ∇ 2 acting on a vector (as opposed toa scalar as in (10.39)), which of course consists of a sum of unit vectors multipliedby components. Two cases arise.(i) If the unit vectors are constants (i.e. they are independent of the values ofthe coordinates) then the differential operator gives a non-zero contributiononly when acting upon the components, the unit vectors being merelymultipliers.356