Q.1 Find the directional derivative of φ(x,y,z)=2x^2+y^2+4xz at (1, 0, 0) in the direction 2i ̂-2j ̂+k ̂
Q.2 Prove that ∇ [r^n ]= nr^(n-2) r ⃗
Q.3 Find div (□((3r) ⃗ )), Where r ⃗=xi ̂+yj ̂+zk ̂ and r^2=x^2+y^2+z^2
Q.4. Varify the Gauss divergence theorem find ∬_s▒〖F ⃗.n ̂ds〗 given that F ⃗=xzi ̂+y^2 j ̂+ yzk ̂ and S is the surface of the cube bounded by planes x=0 , x=1 , y=o, y=1 ,z=0 , z=1
Q.5 A Fluid motion is given by q ⃗=(y+z) i ̂+ (z+x) j ̂+ (x+y)k ̂ find its velocity potential.
Q.6 Use Divergence Theorem to evaluate the integral ∯_S▒〖A.dS〗 where A=x^3 i+y^3 j+z^3 k and S is the surface of spherex^2+y^2+z^2= a^2. .
Q.7 Evaluate∬_s▒〖F ⃗.n ̂ds〗, where F ⃗=zi ̂+xj ̂-3y^2 zk ̂ and S is the surface of the cylinder x^2+y^2=16 included in the first octant between z=0 and z=5
Q.8 Prove that div curl A ⃗ = 0
Q.9 Find ∇^2 [〖2r〗^n ]
Q.10 Use Green’s Theorem to evaluate the integral∮_c▒〖(xy+y^2 )dx+x^2 dy〗 where C is bounded by y = x and y=x^2
Q.11 Find the values of a, b and c such that
A ⃗=(x+2y+az) i ̂+ (bx-3y-z) j ̂+ (4x+cy+2z)(k ) ̂is irrotational vector field. Also, find its scalar potential.
Q.12 Find a unit vector normal to the surface x^2 y+2xz=4 at the point (2,-2,3)
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