. Determine the constants 𝑎 and 𝑏 so that the surface 3𝑥2−2𝑦2−3𝑧2+8=0 is orthogonal to the surface 𝑎𝑥2+𝑦2=𝑏𝑧 at the point (−1, 2,1). 7. If 𝑟⃗=𝑥 𝑖+𝑦𝑗+𝑧𝑘 ̂ and 𝑟=| 𝑟⃗| then show that div (𝑟𝑛𝑟⃗)=(𝑛+3)𝑟𝑛 8. If 𝑟⃗=𝑥 𝑖+𝑦𝑗+𝑧𝑘 ̂ and 𝑟=| 𝑟⃗| then show that 𝑟⃗ 𝑟3 is solenoidal. 9. If 𝑟⃗=𝑥 𝑖+𝑦𝑗+𝑧𝑘 ̂ and 𝑟=| 𝑟⃗| then show that ∇2𝑓(𝑟)=𝑓′′(𝑟)+2 𝑟 𝑓′(𝑟) . 10. If 𝑟⃗=𝑥 𝑖+𝑦𝑗+𝑧𝑘 ̂ and 𝑟=| 𝑟⃗|then show that ∇2(𝑟𝑛+1)=(𝑛+1)(𝑛+2)𝑟𝑛−1. 11. Obtain the directional derivative of 𝜙(𝑥,𝑦)=𝑥3−3𝑥𝑦+4𝑦2 along the vector 𝑢= cos𝜋 6 𝑖+sin𝜋 6 𝑗 and also find 𝐷𝑢𝜙(1,2). 12. Determine the value of the constant ‘a’ such that: 𝐹 ⃗⃗⃗⃗=(𝑎𝑥𝑦−𝑧3)𝑖+(𝑎−2)𝑥2 𝑗+(1−𝑎) 𝑥𝑧2𝑘 ̂ is irrotational and hence find a scalar function 𝜙 such that 𝐹⃗= ∇𝜙 . 13. Show that 𝐹 ⃗⃗⃗⃗=(2𝑥𝑦2+𝑦𝑧)𝑖+(2𝑥2𝑦+𝑥𝑧+2𝑦𝑧2) 𝑗+(2𝑦2𝑧+𝑥𝑦)𝑘 ̂ is a conservative force field. Find its scalar potential. 14. Obtain the curl and divergence of the vector field 𝑭 ⃗⃗⃗=𝑥𝑦𝑧 𝒊+𝑥2𝑦2𝑧 𝒋+𝑦𝑧3𝒌 and hence find scalar potential 𝜙 if 𝑐𝑢𝑟𝑙 𝐹⃗=0. 15. Show that the vector field � �⃗=(𝑥2−𝑦𝑧)𝐢+(𝑦2−𝑥𝑧)𝐣+(𝑧2−𝑥𝑦)𝐤 is irrotational. Obtain the scalar potential 𝜙 such that 𝐅⃗=∇�