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.

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.

Classify the following partial differential equations:(i) 𝜕2𝑢𝜕𝑥2 + 3 𝜕2𝑢𝜕𝑦2 = 𝜕𝑢

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

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

The area bounded by 𝑥=𝑦2−2 and 𝑥=𝑒𝑦  between  𝑦=−1 and 𝑦=1 is revolved about the line 𝑦=1. The Volume integral for the solid generated is