Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUS∇(φ + ψ) =∇φ + ∇ψ∇ · (a + b) =∇ · a + ∇ · b∇×(a + b) =∇×a + ∇×b∇(φψ) =φ∇ψ + ψ∇φ∇(a · b) =a × (∇×b)+b × (∇×a)+(a · ∇)b +(b · ∇)a∇ · (φa) =φ∇ · a + a · ∇φ∇ · (a × b) =b · (∇×a) − a · (∇×b)∇×(φa) =∇φ × a + φ∇×a∇×(a × b) =a(∇ · b) − b(∇ · a)+(b · ∇)a − (a · ∇)bTable 10.1 Vector operators acting on sums and products. The operator ∇ isdefined in (10.25); φ and ψ are scalar fields, a and b are vector fields.Therefore the curl of the velocity field is a vector equal to twice the angularvelocity vector of the rigid body about its axis of rotation. We give a fullgeometrical discussion of the curl of a vector in the next chapter.10.8 Vector operator formulaeIn the same way as for ordinary vectors (chapter 7), for vector operators certainidentities exist. In addition, we must consider various relations involving theaction of vector operators on sums and products of scalar and vector fields. Someof these relations have been mentioned earlier, but we list all the most importantones here for convenience. The validity of these relations may be easily verifiedby direct calculation (a quick method of deriving them using tensor notation isgiven in chapter 26).Although some of the following vector relations are expressed in Cartesiancoordinates, it may be proved that they are all independent of the choice ofcoordinate system. This is to be expected since grad, div and curl all have cleargeometrical definitions, which are discussed more fully in the next chapter andwhich do not rely on any particular choice of coordinate system.10.8.1 Vector operators acting on sums and productsLet φ and ψ be scalar fields and a and b be vector fields. Assuming these fieldsare differentiable, the action of grad, div and curl on various sums and productsof them is presented in table 10.1.These relations can be proved by direct calculation.354