Mathematical Methods for Physics and Engineering - Matematica.NET

LINE, SURFACE AND VOLUME INTEGRALSyCRdxdrdyˆn dsxFigure 11.11 A closed curve C in the xy-plane bounding a region R. Vectorstangent and normal to the curve at a given point are also shown.The surface integral over S 1 is easily evaluated. Remembering that the normal to thesurface points outward from the volume, a surface element on S 1 is simply dS = −k dx dy.On S 1 we also have a =(y − x) i + x 2 k,sothat∫ ∫∫I = − a · dS = x 2 dx dy,S 1 Rwhere R is the circular region in the xy-plane given by x 2 + y 2 ≤ a 2 . Transforming to planepolar coordinates we have∫∫∫ 2π ∫ aI = ρ 2 cos 2 φ ρ dρ dφ = cos 2 φdφ ρ 3 dρ = πa4R ′ 00 4 . ◭It is also interesting to consider the two-dimensional version of the divergencetheorem. As an example, let us consider a two-dimensional planar region R inthe xy-plane bounded by some closed curve C (see figure 11.11). At any pointon the curve the vector dr = dx i + dy j is a tangent to the curve and the vectorˆn ds = dy i − dx j is a normal pointing out of the region R. If the vector field a iscontinuous and differentiable in R then the two-dimensional divergence theoremin Cartesian coordinates gives∫∫R( ∂ax∂x + ∂a y∂y)∮dx dy =∮a · ˆn ds = (a x dy − a y dx).CLetting P = −a y and Q = a x , we recover Green’s theorem in a plane, which wasdiscussed in section 11.3.11.8.1 Green’s theoremsConsider two scalar functions φ and ψ that are continuous and differentiable insome volume V bounded by a surface S. Applying the divergence theorem to the402