082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

26.3 The gradient of a scalar field 851
3 (a) 2xi+k (b) i
4
5
(c) −j +2k
(d) 2i
(e) 0 (f) 0
∂v
∂x =3yzi+2xj+k
∂v
= 3xzi −2yj +2yk
∂y
∂v
= 3xyi +2zj
∂z
∂ 2 v
∂x 2 = 2j
∂ 2 v
∂y 2 =−2j+2k
∂ 2 v
∂z 2 = 2j
∂v
∂x =yzcos(xyz)i +yzexy j −2yk
∂v
∂y =xzcos(xyz)i +xzexy j −2xk
∂v
∂z =xycos(xyz)i +exy j
6 φv =x 2 yzi+x 3 y 2 zj−3x 4 yzk
7
∂(φv)
= 2xyzi +3x 2 y 2 zj −12x 3 yzk
∂x
∂φ
∂x =yz
∂v
∂x =i+2xyj−9x2 k
∂v
∂x = i x +2ycoszj −4x3 yzk
∂v
∂y = i y +2xcoszj−x4 zk
∂v
∂z = −2xysinzj −x4 yk
∂ 2 v
∂x 2 = − i
x 2 −12x2 yzk
∂ 2 v
∂y 2 = − i
y 2
∂ 2 v
∂z 2 = −2xycoszj
26.3 THEGRADIENTOFASCALARFIELD
Given a scalar functionofx,y,z
φ = φ(x,y,z)
we can differentiate it partially w.r.t. each of its independent variables to find ∂φ
∂x , ∂φ
∂y
and ∂φ . If wedo this, the vector
∂z
∂φ
∂x i + ∂φ
∂y j + ∂φ
∂z k
turns out to be particularly important. We call this vector the gradient of φ and denote
itby
∇φ or grad φ
An alternative form of writing ∇φ isas threecomponents
( ∂φ
∂x , ∂φ
∂y , ∂φ )
∂z
gradφ=∇φ= ∂φ
∂x i + ∂φ
∂y j + ∂φ
∂z k
The process of forming a gradient applies only to a scalar field and the result is always
a vector field.