154CHAPTER 2. CURVILINEAR COORDINATES(2.3) Prove that each of the following systemsa) Parabolic cylindrical coordinates (u, v, z) defined byx = u v, y = [u? – u²], z = zb) Paraboloidal coordinate system (u, v, 4) defined byI = UV COS¢, y = uv sin o, z =(u² – v]%3Dis orthogonal, and then find the following for each of them:The scale factors, the base unit vectors, the vector de, the elementde? and the element of volume dr.[hint: to show that the system is orthogonal prove that h12h23 = 0]h13Gradient, Divergence, Laplacian and Curl

The given question can be solved as shown in step2.

(2.3) Prove that each of the following systems a) Parabolic cylindrical coordinates (u, v, z) defined by * = uv, y = (u? – u²], z = z b) Paraboloidal coordinate system (u, v, ) defined by I = uv cos ø, · y = uv sin ø, z = [u? – v²] is orthogonal, and then find the following for each of them: The scale factors, the base unit vectors, the vector de, the element de? and the element of volume dr. [hint: to show that the system is orthogonal prove that h12 h23 = 0] h13 %3D %3D