General orthonormal curvelinear coordinates (u, v, w) can be obtained from cartesian coordinates by the transformation →x = →x(u, v, w). The unit vectors are then given by: →eu = 1 h1 ∂→x ∂u , →ev = 1 h2∂→x ∂v , →ew = 1 h3 ∂→x ∂w where the factors hi set the norm to 1. Then holds: gradf = 1 h1 ∂f ∂u→eu + 1 h2.. By itself the del operator is meaningless, but when it premultiplies a scalar function, the gradient operation is defined. We will soon see that the dot and cross products between the del operator and a vector also define useful operations. With these definitions, the change in f of (3) can be written as. df = ∇f ⋅ dl = | ∇f | dlcosθ.
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The result of cross-multiplying A by the del operator, defined by (2.1.6), is the curl operator. This is the reason for the alternate notation for the curl operator. Thus, in Cartesian coordinates The problems give the opportunity to derive expressions having similar forms in cylindrical and spherical coordinates. The results are summarized in.. Del in cylindrical and spherical coordinates Table with the del operator in cylindrical and spherical coordinates Operaion Cartesian coordinates (x,y,z) Cylindrical coordinates (ρ,φ,z) Spherical coordinates (r,θ,φ) Definition of coordinates x = ρcosφ y = ρsinφ z = z x = rsinθcosφ y = rsinθsinφ z = rcosθ ρ = p x2 +y2
