Hi everyone! I have a question about mathematical methods in physics. The question is Turkish in attached but I translated it for you. Can you help me please? Thx a lot. (a) Show that the function ω satisfies the condition ∇⃗ u ∙ ∇⃗ v = 0 so that the function is analytical (where ∇⃗ is the del operator and it is defined as ∇⃗ ∶ = î × ∂ / ∂x + î × ∂ / ∂y. U = î × ∂u / ∂x + î × ∂u / ∂y. Similarly, ∇⃗ v is defined.)

(b) Show that the function ω must be analytical to be a harmonic function, that is, it must satisfy the Cauchy-Riemann conditions. (Hint: If a function u (x, y) is harmonic, it satisfies the following condition: ∇ ^ 2u = ∂ ^ 2u / ∂x ^ 2 + ∂ ^ 2u / ∂y ^ 2 = 0. In this case, u and v must meet the harmonics condition separately.)

a solenoid is created by wrapping a l = 90 m long wire around a hollow tube of diameter d = 4.5 cm.   the wire diameter is d = 0.9 mm.   the solenoid wire is then connected to a power supply so that a current of i = 9 a flows through the wire.

randomized variablesl = 90 m

d = 4.5 cm

d = 0.9 mm

i = 9 a                                          

write an expression for the number of turns, n, in the solenoid. you do not need to take into account the diameter of the wire in this calculation.

calculate the number of turns, n, in the solenoid.

write an expression for the length of the solenoid (l2) in terms of the diameter of the hollow tube d.

calculate the length of the solenoid (l2)   in meters.

calculate the magnitude of the magnetic field at the center of the solenoid in teslas.

explanation:

the temperature and the pressure affect solubility. temperature affects by changing the properties of the solvent. the increase in pressure increases solubility, the decrease in pressure decreases solubility.