If v = 3x2yi + 2xyzj − 3x4y2k, find ∂v∂x , ∂v∂y , ∂v∂z . Further, find ∂2v∂x2 and ∂2v∂x∂z .Solution We find∂v∂x = 6xyi + 2yzj − 12x3y2k∂v∂y = 3x2i + 2xzj − 6x4yk∂v∂z = 2xyj∂2v∂x2 = 6yi − 36x2y2k∂2v∂x∂z = 2yjEXERCISES 26.21 Given v = 2xi + 3yzj + 5xz2k find(a) ∂v∂x (b) ∂v∂y (c) ∂v∂z2 If f = 2i − xyzj + 3x2zk find(a) ∂f∂x (b) ∂f∂y (c) ∂f∂z(d) ∂2f∂x2 (e) ∂2f∂y2 (f) ∂2f∂z23 Given E = (x2 + y)i + (1 − z)j + (x + 2z)k find(a) ∂E∂x (b) ∂E∂y (c) ∂E∂z(d) ∂2E∂x2 (e) ∂2E∂y2 (f) ∂2E∂z24 If v = 3xyzi + (x2 − y2 + z2 )j + (x + y2 )k find∂v∂x , ∂v∂y , ∂v∂z , ∂2v∂x2 , ∂2v∂y2 , and ∂2v∂z2 .5 If v = sin(xyz)i + z exyj − 2xyk find∂v∂x , ∂v∂y , ∂v∂z .6 If v = xi + x2yj − 3x3k, and φ = xyz, findφv, ∂∂x (φv), ∂φ∂x , ∂v∂x . Deduce that∂∂x (φv) = φ ∂v∂x + ∂φ∂x v7 If v = ln(xy)i + 2xy cos zj − x4yzk, find∂v∂x , ∂v∂y , ∂v∂z , ∂2v∂x2 , ∂2v∂y2 , and ∂2v∂z2 .Solutions1 (a) 2i + 5z2k (b) 3zj(c) 3yj + 10xzk2 (a) −yzj + 6xzk (b) −xzj(c) −xyj + 3x2k (d) 6zk(e) 0 (f) 0Downloaded From :www.EasyEngineering.netDownloaded From :www.EasyEngineering.netwww.EasyEngineering.net