If 𝑣 = 𝑟𝑚, where 𝑟2 = 𝑥2 + 𝑦2 + 𝑧2, then prove that 𝑣𝑥𝑥 + 𝑣𝑦𝑦 + 𝑣𝑧𝑧 = 𝑚(𝑚 + 1)𝑟𝑚−2.

If 𝑣 = 𝑟𝑚, where 𝑟2 = 𝑥2 + 𝑦2 + 𝑧2, then prove that 𝑣𝑥𝑥 + 𝑣𝑦𝑦 + 𝑣𝑧𝑧 = 𝑚(𝑚 + 1)𝑟𝑚−2.

If 𝑢 = 𝑥2 tan−1 𝑦𝑥 − 𝑦2 tan−1 𝑥𝑦 show that 𝜕2𝑢𝜕𝑥𝜕𝑦 = 𝑥2−𝑦2𝑥2+𝑦2 and 𝜕2𝑢𝜕𝑥𝜕𝑦 = 𝜕2𝑢𝜕𝑦𝜕𝑥.

7. Find the gradient for the function 𝑢 = 𝜑(𝑟), 𝑟 = √𝑥2 + 𝑦2 + 𝑧2.Answer: 𝛁𝑢 = 𝜑′(𝑟) 𝒓𝑟 .

Evaluate the integral: ∫12∫02𝑥−𝑥2(𝑥2+𝑦2)−1/2𝑑𝑦𝑑𝑥Group of answer choices2+ln⁡(2+1)ln⁡|sec⁡𝜃+tan⁡𝜃|+𝐶2−ln⁡(2+1)ln⁡(2−1) PreviousNext

If x,y,z∈R, x+y+z=5, x2+y2+z2=9𝑥,𝑦,𝑧∈𝑅, 𝑥+𝑦+𝑧=5, 𝑥2+𝑦2+𝑧2=9, then length of interval in which x𝑥 lies is