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

832 Chapter 25 Functions of several variables
Solutions
1
2 (a)
∂ 2 v
∂h 2 = − r2
4h 3/2 , ∂2 v
∂h∂r = r √
h
, ∂2 v
∂r 2 = 2√ h
∂ 2 f
∂x 2 = 2y, ∂2 f
∂x∂y = 2x, ∂2 f
∂y 2 = 6y
(b) 24x 2 y 3 −18xy 5 ,
24x 3 y 2 −45x 2 y 4 ,
12x 4 y −60x 3 y 3
(c) −x −3/2 y 2 ,4x −1/2 y,8 √ x
(d) 2 y , −2x y 2 , 2(x2 +1)
y 3
(e) 18xy −1/2 , − 9 2 x2 y −3/2 , 9 4 x3 y −5/2
(f) −x −3/2 y 1/2 ,x −1/2 y −1/2 , −x 1/2 y −3/2
∂ 2 z
3 (a)
∂x 2 = 0, ∂2 z
∂x∂y =2e2y , ∂2 z
∂y 2 = 4xe2y
(b) −2y 2 sin(xy),
4 (a)
2cos(xy) −2xysin(xy),
−2x 2 sin(xy)
(c) −4sin(2x +3y) −4xcos(2x +3y),
−3sin(2x +3y) −6xcos(2x +3y),
−9xcos(2x +3y)
(d) −16y 3 sin(4xy),
8ycos(4xy) −16xy 2 sin(4xy),
8xcos(4xy) −16x 2 ysin(4xy)
(e) e x siny,e x cosy, −e x siny
(f) 9e 3x−y ,−3e 3x−y ,e 3x−y
(g) y 2 e xy ,e xy (1 +xy),x 2 e xy
∂ 2 z
∂x 2 = 3420(3x −2y)18 ,
∂ 2 z
∂x∂y = −2280(3x −2y)18 ,
5
∂ 2 z
= 1520(3x −2y)18
∂y2 (b) −(2x +5y) −3/2 ,
− 5 2 (2x +5y)−3/2 ,
− 25 (2x +5y)−3/2
4
(c) 2cos(x 2 +y 2 ) −4x 2 sin(x 2 +y 2 ),
−4xysin(x 2 +y 2 ),
2cos(x 2 +y 2 ) −4y 2 sin(x 2 +y 2 )
4
(d) −
(2x +5y) 2,
(e)
10
−
(2x +5y) 2,
25
−
(2x +5y) 2
18
(3x −2y) 3 , − 12
(3x −2y) 3 , 8
(3x −2y) 3
∂ 3 z
∂x 3 = 0, ∂ 3 z
∂x 2 ∂y = − 2
(y +1) 2,
∂ 3 z
∂x∂y 2 = 4x ∂3 z
(y +1) 3, ∂y 3 = − 6x2
(y +1) 4
6 (a)
∂ 2 f
(2,1) = 7.3891,
∂x2 ∂ 2 f
(2,1) = 7.3891,
∂x∂y
∂ 2 f
∂y 2 (2,1)=0
(b) −0.5645,0.2822, −0.1411
(c) 0.25,0, −1
25.5 PARTIALDIFFERENTIALEQUATIONS
Partial differential equations (p.d.e.s) occur in many areas of engineering. If a variable
dependsupontwoormoreindependentvariables,thenitislikelythisdependencecanbe
describedbyap.d.e.Theindependentvariablesareoftentime,t,andspacecoordinates
x,y,z.
One example is the wave equation. The displacement,u, of the wave depends upon
time and position. Under certain assumptions, the displacement of a wave travelling in
one direction satisfies
∂ 2 u ∂2 u
∂t 2 =c2 ∂x 2