Advanced Calculus fi..

Chapter 3 Vector Differential Calculus 195The three terms on the right are simply the three terms of the Laplacianat the origin.PROBLEMS1. Prove (3.48). [Hint: Use the result of Problem 5 following Section 2.12.12. Prove (3.47). [Hint: Use (3.48) and the rules (1.19) of Section 1.2.13. Prove (3.46). [Hint: Use (3.48) and the rules (1.19) of Section 1.2.14. Prove that when the vectors V F, VG, V H are mutually perpendicular in D, the coordinatesare orthogonal.5. Prove (3.62). [Hint: Use (3.56) to write:( 1 ar) ( 1 ar) ( 1 ar)p=(upu) -- + BP~ -- +(YP",)a2 a~ p2 a~--y2 a~ -tUse (3.28) and (3.55) to show thatcurl p = grad (up,) x +grad(B~u) X -p(2).av ....To compute [curl p],, take the scalar product of both sides with and use (3.59) and(3.60) to compute the scalar triple products.]6. Verify the following relations for cylindrical coordinates u = r, v = 0, w = z:a) the surfaces r = const, 0 = const, z = const form a triply orthogonal family, andb) the element of arc length is given byds2 = dr2 + r2de2 + dz2;c) the components of a vector p are given byp, =pxcosO+pysinO, pe =-pxsinO+pycose, Pz=Pz;au i au aud) grad U has components: T, f ;?B, x;r) ciiv p = f [$(rp,) + % + r t];f) curl p has components:g) V ~ U is given by (3.66).7. Verify the following relations for spherical coordinates u = p, v = 4, w = 8:a) the surfaces p = const, 4 =const, 0 = const form a triply orthogonal family, andJ = a(x, Y, Z) = p2 sin 4;a(p, 60)b) the element of arc length is given byds2 = dp2 + p2&p2 + p2 sin2 I$do2;