Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 1
ASSIGNMENT C
1).Compute 2 2 2
2
2 2, , if ln
z z zz x y
x x y y
2. Find 2 2 2
2 2
2 2, , if , , ,
z z zz f u v u x y v xy
x yx y .
3.) Verify that if
y
xz z
x y xy z z xy xex y
.
4.) Show that if 2 2 2
2 2 22 2 2
1, 0
u u uu
x y zx y z .
5.) Find the derivative of the function 2 2lnz x y at point P (1, 1) and following the bisectrix in the first
quadrant.
6.) Find the Limits of the double integral ,D
f x y dxdy over the region D.
a.) Triangle with sides x=0, y=0, y + x=2
b.) 2 2, 4y x y x
7.) Compute 2 2
2 2
1
1D
x y
x y where the domain D is specified by the inequalities
2 2 1, 0, 0x y x y
8.) Express the triple integral , ,f x y z dxdydz in cylindrical coordinates where is the region bound by
thecylinder2 2 2x y x ,the plane z=0 and parabolic
2 2z x y .
9.) Find the centric of the tetrahedron in the first octant enclosed by the coordinate planes and the plane
1x y z .
10.) Find the volume of the solid that enclosed between the parabolic 2 2 3 28 and 3z x y z x y .
11.) Find the volume of the solid that enclosed between the sphere 2 2 2 22x y z a and parabolic
2 2 , 0.az x y a
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 2
SOLUTION FOR ASSIGNMENT C
1).Compute 2 2 2
2
2 2, , if ln
z z zz x y
x yx y
2
2
2 2
2
2 2
2
22
2 2
22
2
22
ln
2
2( ) ( )
2( ) 2 (2 )
2 2 4
2( )
x yz
x x
x y
xx
x y x y
z z x
x x x x x y
x y x x
x y
x y x
x y
y x
x y
2
2ln
z
x y
x yz
y y
2
2
2
1
x y
y
x y
x y
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 3
2
2
22
2
22
1( )
2
2
z z
x y x y x x y
x
x y
z x
x y x y
2
2
2
2 2
22
2
2 22
1
1
1
z
y
z z
y y yy x y
x y
zso
y x y
2
222
1z
y x y,
2
22
2z x
x y x y,
2 2
2 22
2( )z y x
x x y
2. Find 2 2 2
2 2
2 2, , if , , ,
z z zz f u v u x y v xy
x yx y
2 2, , ,z f u v u x y v xy
Chain Rule
z dz u dz v
x du x dv x
z dz u dz v
y du y dv y
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 4
2 2( ) ( )2 ( )
2
u v
x u v
z dz x y dz xyz x z y
x du x dv x
z xz yz
2 2( ) ( )2 ( )
2
u v
y u v
z dz x y dz xyz y z x
y du y dv y
z yz xz
2
2
z
x
2
2( ) ( )
(2 ) 2 2
x
u v u ux vx
z zz
x x xx
xz yz z xz yzx
2
22u ux vx
zz xz yz
x
2 z
xdy
2
2( ) ( )
(2 ) 2
x
u v ux v vx
z zz
x y xx
yz xz yz z xzx
2
2 ux v vx
zyz z xz
x y
2
2
z
y
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 5
2
2( ) ( )
(2 ) 2 2
y
u v u uy vy
z zz
y y yy
yz xz z yz xzy
2
22 2u uy vy
zz yz xz
x
3.) Verify that if
y
xz z
x y xy z z xy xex y
y
xz xy xe
2
y
x
y y
x x
y
x
y
x
y
x
y
x
y
x
y y
x x
y y
x x
y
zxy xe
x x
yy e e
x
x yy e
x
z x yy e
x x
zxy xe
y y
yx xe
y x
zx e
y
z z x yx y x y e y x e
x y x
xy x y e yx ye
xy xe
( )
y y
x x x
y
x
ye ye
xy xy xe
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 6
y
xz xy xe
z zx y xy z
x y
z zx y xy z
x y
4.) Show that if 2 2 2
2 2 22 2 2
1, 0
u u uu
x y zx y z
2 2 2
2 2 2
1u
x y z
v x y z
3 3
2 2 22 2
3
2
1
1 1( )
2 2
x
x x
uv
v v x y z v
xv
3
2
3 3
2 2
3 5
2 2
3 5
2 2 22 2
3 5
22 2
3
2
3
2
3
xx
x
x
x
u xv
v x v
v xv v
v x x y z v
v x v
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 7
3 3
2 2 22 2
3
2
1
1 1( )
2 2
y
y y
uv
v v x y z v
yv
3
2
3 3
2 2
3 5
2 2
3 5
2 2 22 2
3 5
22 2
3
2
3
2
3
yy
y
y
y
u yv
v y v
v yv v
v y x y z v
v y v
3 3
2 2 22 2
3
2
1
1 1( )
2 2
z
z z
uv
v v x y z v
zv
3
2
3 3
2 2
3 5
2 2
3 5
2 2 22 2
3 5
22 2
3
2
3
2
3
zz
z
z
z
u zv
v z v
v zv v
v z x y z v
v z v
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 8
3 5
22 2
3 5
22 2
3 5
22 2
3 52 2 22 2 22 2
2 2 2
3
3
3
____________________________________
3 3
xx
yy
zz
u v x v
u v y v
u v z v
u u uv x y z v
x y z
3 52 2 2
2 22 2 2
3 51
2 2
3 3
2 2
2 2 2
2 2 2
3 3
3 3
3 3
0
u u uv vv
x y z
v v
v v
u u u
x y z
2 2 2
2 2 20
u u u
x y z
5.) Find the derivative of the function 2 2lnz x y at point P (1, 1) and following the bisectrix in the first
quadrant.
z P (1, 1)
2 2 2 2
2 2
2 2
2 2
1ln ln
2
2
(1,1) 1
x
x
z x y x y
d x y
dx
x y
x
x y
z
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 9
2 2 2 2
2 2
2 2
2 2
1ln ln
2
2
1,1 1
y
y
z x y x y
d x y
dy
x y
y
x y
z
1 1 2p
1
2
2 cos 14
2 sin 14
u
u
u i j
P
2
2v i j
2 2
2 2vD z i j
2 2
2 2vD z i j
6.) Find the Limits of the double integral ,D
f x y dxdy over the region D.
a.) Triangle with sides x=0, y=0, y + x=2
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 10
20 2,0 xx y
2 }{( , ),0 2,0 xD x y x y
2 }{( , ),0 2,0 xD x y x y
b.) 2 2, 4y x y x
2
22
4
y xx
y x
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 11
2 2{( , ), 2 2, 4 }D x y x x y x
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 12
7.) Compute 2 2
2 2
1
1D
x y
x y where the domain D is specified by the inequalities 2 2 1, 0, 0x y x y
2{( , ), 0 1,0 1 } {( , ),0 ,0 1}2
D x y x y x or D r r
1 1 1 1 12 2 2 32 2
2 4 4 4 40 0 0 0 0 0 0
1 12 4
4 40 0
2 4
1 1
2 21 1 1 1 1
( ) (1 )
2 1 1
11 1arcsin 1
02 2 2
1 1arcsin 1
2 2 2
arcsin1 14
r r r r rdr r drrdrd rdrd dr
r r r r r
d r d r
r r
r r
21y x
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 13
2 2
2 2
1arcsin1 1
41D
x y
x y
8.) Express the triple integral , ,f x y z dxdydz in cylindrical coordinates where is the region bound by
thecylinder 2 2 2x y x ,the plane z=0 and parabolic 2 2z x y .
2 2 2x y x 2 2( 1) 1x y (1, 0) r=1
2 2
2 2
22
x y xz x
z x y 2 cosz r
2 2 2 2 cos
2cos
r x y r
r
2, , :0 2 ,1 2cos ,0r z r z r
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 14
22 2cos
0 1 0
2 2cos3
0 1
42
0
2 24
0 0
2 22 2
0 0
, ,
2cos
14
4 (cos )
4 (cos )
r
f x y z dxdydz rdzdrd
r drd
rd
d d
d d
2
2 22
0 0
2 22
0 0
2 2
0 0
2 2 2 2
0 0 0 0
1 cos 2cos
2
1 cos 24 ( )
2
(1 2cos 2 cos 2 )
1 cos 41 2cos 2
2 2
3 1cos 2 (2 ) cos 4 (4 )
2 8
213 sin 2 sin 4
08
3 2
d d
d d
d d
d d d d
, ,f x y z dxdydz
2, , :0 2 ,0 2cos ,0r z r z r
, ,f x y z dxdydz
9.) Find the centric of the tetrahedron in the first octant enclosed by the coordinate planes and the plane
1x y z .
Norton University [MATHEMATICS FOR ENGINEERING III
Department: Civil Engineering Page 15
1
1x y z (k)
1 1 1
0 0 0
1 1
0 0
12
0
3 2
1
1 1
2 2
11 1 1
06 2 2
6
x x y
x
m kdzdydx
k x y dydx
k x x dx
k x x
km unit mass
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Department: Civil Engineering Page 16
1 1 1
0 0 0
1 12
0 0
31
2
0
2 3 4
( )
2 2
1
04 3 8
24
x x y
yz
x
M k xdzdydx
k x x xy dydx
x xk x dx
x x xk
kunit of moment
1 1 1
0 0 0
1 12
0 0
31
0
31
0
4
(1 )2
16
1 16
11
024
24
x x y
yz
x
M k zdzdydx
kx y dydx
kx dx
kx d x
kx
kunit of moment
1 1 1
0 0 0
1 1
0 0
1 12
0 0
31
0
31
0
4
(1 )
( 1 )
16
1 16
11
024
24
x x y
xz
x
x
M k ydzdydx
k x y ydydx
k y x y dydx
kx dx
kx d x
kx
kunit of moment
Center of mass_ _ _
( , , )x y z_ _ _
, ,yz xyxz
M MMx y z
m m m
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Department: Civil Engineering Page 17
_
yzMx
m
124
4
6
k
k
_xzM
ym
124
4
6
k
k
_xyM
zm
124
4
6
k
k
Centric of mass _ _ _1 1 1
, ,4 4 4
x y z
10.) Find the volume of the solid that enclosed between the parabolic 2 2 2 28 andz x y z x y .
2 2
2 2
2 24
8
z x yx y
z x y 2r
2 2z x y
2 28z x y
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Department: Civil Engineering Page 18
2 2 2 2, , : 2 2, 2 2, 8V x y z x y x y z x y
2 2, , : 0 2 ,0 2, 8V r z r r z r
2
2
2 2 8
0 0
2 2 2 22 3
0 0 0 0
22 4
0
2
0
8 2 8 2
214
02
8
16
r
solidr
V rdzdrd
r rdrd r r drd
r r d
d
unit of volume
16solidV unit of volume
11.) Find the volume of the solid that enclosed between the sphere 2 2 2 22x y z a and parabolic
2 2 , 0.az x y a
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Department: Civil Engineering Page 19
2 2
2
2 2
0 0
2 22 2
0 0
22 2
0
32 2
0 0
32 2 2 2
0 0
42 2 3
3 3
( 2 )
2 ( 2 )
2 2
12 2 2
2
12 (2 )
03 4
1 1 2 22
3 4
a a r
solid
r
a
a
a
a a
a a
V rdzd dr
ra r rd dr
a
ra r rdr
a
ra r rdr dr
a
ra r d a r dr
a
ara r
a
a a 3
3 3
3
03
7 8 22
12 12
8 2 76
a
a a
a
3 8 2 76
solidV a unit of volume