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

876 Chapter 27 Line integrals and multiple integrals
Inthree dimensions, if
F =P(x,y,z)i +Q(x,y,z)j +R(x,y,z)k
isconservative, wecan write
So
F=∇φ
= ∂φ
∂x i + ∂φ
∂y j + ∂φ
∂z k
P = ∂φ
∂x
Q = ∂φ
∂y
R = ∂φ
∂z
You should refer back toExample 26.4 where thisidea was introduced.
Example27.6 Thevector field visderivable from the potential φ = 2xy +zx. Find v.
Solution If v isderivable from the potential φ, then v = ∇φ and so
v=∇φ=(2y+z)i+2xj+xk
This vector field isconservative asiseasilyverified by findingcurl v. Infact,
i jk
∇×v=
∂ ∂ ∂
∂x ∂y ∂z
∣
2y+z2x x
∣
=0i−(1−1)j+(2−2)k
= 0
Indeed, recall from Example 26.10 thatcurl (grad φ) isidentically zero forany φ.
When the vector field F is conservative there is an alternative method of evaluating the
line integral ∫ F·ds which involves the use of the potential function φ. Consider the
C
following example.
Example27.7 Thetwo-dimensional vector fieldF = i +2j isconservative.
(a) Findasuitablepotential function φ suchthat F = ∇φ = ∂φ
∂x i + ∂φ
∂y j.
(b) Evaluate ∫ (3,2)
F·ds along any convenient path.
(0,0)
(c) Find the value of φ at B(3, 2), and the value of φ at A(0, 0), and show that the
differencebetweentheseisequaltothevalueofthelineintegralobtainedinpart(b).
Solution (a) We are given that ∂φ = 1 so that φ = x + f (y), where f is an arbitrary function
∂x
ofy. We are also given that ∂φ = 2 so that φ = 2y +g(x), wheregis an arbitrary
∂y
function ofx. Itiseasy toverify that φ =x +2yisasuitablepotential function.