Problem 4: A non Cartesian system is called a curvilinear system. Starting from Cartesian variables {z, y, z), curvilinear variables (v₁, U2, U3} can be introduced from Cartesian variables by relating each of z. y, and 2 to 0₁, 02, and v3. This yields defining h₁ = h₂ = h3 = Ər Vavr VV 2 2 x = a cosh μ cos , + 2 2 2 Ər VƏv₂, - (35) ² + + ду Əv₁ + ду Əv₂ Ər Vvs, as the metric coefficients. The gradient of a scalar V in a curvilinear system can be found using the metric coefficients as v₁ + 2 Əy მავ - (35) ² 2 + + 1 ᎧᏙ h₁ Əv₁ 1 ᎧᏙ 1 ƏV v₂ + h₂ Əv₂ hვ მ3. where V₁, V2, and v3 are the associated curvilinear bases. As an example, obtain the gradient in elliptical cylindrical coordinates {μ, o, z} defined by Əz 2 y=asinh | sin ¢, where a is a constant. Hint: (cosh z)' = sinhz, and (sinh x)' = coshx. V3, 2=2},

Fig: 1