Vector Calculus Tutorial and Solution
TE 2002-EngrgMaths I
(2006/2007)
[SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE Department of Electrical and Computer Engineering
TE 2002 - ENGINEERING MATHEMATICS I TUTORIAL 1 1.
The points A, B, C have coordinates (3,2,6), (4,-1,2), (-2,3,7), referred respectively to rectangular cartesian axes Oxyz. Calculate AB x AC and deduce the area of the triangle ABC.
2. A curve is represented in parametric form by the vector equation
r(t) = (2t + 1) i
? + (t - 2) j ?
Find the equation of the curve.
3. If r = r(t), show that at t = 0
r
r r r t r dt d &&&&&.2=+ .
Hint: Note that |V |2 = V . V
4i). Find the unit tangent 21t and t to the parametric line v = constant, u = constant at the point P(1,1,1) on the surface
x =
u
y = v (u, v positive)
z = 1/uv
ii) Find a vector normal to the surface at the point Q (u = 1/2, v = 1/3 )
iii) Find the equation of the tangent plane at the point Q.
5. The temperature at the point (x,y,z) at time t is T(x,y,z,t) = xy 2 + 2yzt + sin xt.
Find the rate of change of temperature, with respect to time, encountered by a particle passing the point (2,3,1) with velocity k j i
V ?2???+= at time t = 0.
6i). Show that
2][])()()([c b a b x a a x c c x b =
ii)
Given the set of vectors
J
b x a C J a x
c B J c x b A ===,, where
0][≠=c b a J ,
show that 1.= (b)
0.=b A (c) J
C B A 1][= (d) If c b a ,, are non coplanar then so are C B A ,,
The vector C B A ,, are known as the reciprocal set of vectors of the vectors
c b a ,,.
7. Practice
Edwards & Penny : Problems sets 12.2 and 12.3
Kreyszig: Problems sec 8.2
Adams: Exercises 7.1
Ans: 1. 11.81
2. x = 2y + 5
4i) )1,0,1(2
11?=t ii) 12j ? + 18j ? + k ?
iii) 12(x-1) +18(y-1) +(z-1) = 0
5. 29=dt
dT
SJC/cc:10.1.2002
tgT1:ms6
TE 2002-EngrgMaths I
(2006/2007)
[SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE Department of Electrical and Computer Engineering
TE 2002 - ENGINEERING MATHEMATICS I
TUTORIAL 2
1a) Find the equation of the line in space through the point (3, -1, 2) in the direction
k j i ?4?3?2+?
b) In what direction does the following line point
x = -3t + 2
y = -2(t - 1)
z = 8t + 2
2. Find the unit vector tangential at any point on the curve
x = t 2 + 1
y = 4t - 3
z = 2t 2 - 6t
Also find the unit tangential vector at the point t = 2.
3. Let the position of a particle at time t be given by
R = e t sint i )+ e t cost j ? - 2k ?
Determine the following
(i) velocity,
(ii) speed,
(iii) acceleration,
(iv) the unit tangent vector, T,
(v) the radius of curvature ρ,
(vi) the unit normal vector n ?
(vii) the normal component of acceleration, a n and
(viii) the tangential component of acceleration a t .
4. A curve is represented by r(t) = 2t i ? + (t + 1)j ?. Find the arc length S from t = 0 to t. Hence parametrize r(t) in terms of arc length.
ENGINEERING MATHEMATICS I; TUTORIAL 2
5. A curve is represented by r(t) = t i ? + e -t j ? . Find the following:
(i) tangent vector T, and
(ii) the radius of curvature ρ or curvature ρ1
=k .
6. Practice
Edward & Penny: Problem sets 12.4, 12.5and 12.6
Kreyzig: Problems Secs. 8.4 & 8.5
Adams: Exercises 7.5
Ans: 1a) (3, -1, 2) + λ(2, -3, 4)
b) (-3, -2, 8)
2. )??2?
2(31
,)13125(?)32(?2?2k j j t t k t j i t +++??++
3. (i) V = [e t (sint + cost), e t (cost - sint), 0]
(ii) V = e t √2
(iii) a = [2e t cost, -2e t sint, 0]
(iv) }0,)sin (cos ,)cos (sin {22
t t t t I ?+=
(v) ρ = √2e t
4. j s
i s s r t s ?)15(?52)(,5++==?
5. 2
32)1(1t t
e e ??+=ρ
SJC/cc:10.1.02
tgT2:ms6
TE2002-EngrgMaths I
(2006/2007)
[SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE Department of Electrical and Computer Engineering
TE 2002 - ENGINEERING MATHEMATICS I
TUTORIAL 3
1.
Find the arc length of (cost, sint, t 2), 0 ≤ t ≤ π Note: c a x x a a x x dx a x +++++=+∫)]([ln [2
12222222
(Ans: 10.63)
2. p = f(u, v, w) = u 2 + v 2 - w
u = x 2y, v = y 2, w = e -xz .
Find x
p ??. (Ans: (2x 2y) (2xy) + ze -XZ )
3. A function f(x, y) has, at the point (1,3) directional derivatives of +2 in direction
toward (2, 3) and -2 in the direction toward (1, 4). Determine the gradient vector at (1, 3) and compute the directional derivative in the direction toward (3, 6).
(Ans: 13
2;)2,2()3,1(??=?f )
4. Compute the equation of the plane tangent to the surface 3xy + z 2 = 4 at (1,1,1)
(Ans: 3x + 3y + 2z = 8)
5. Consider an elliptic paraboloid z = c - ax 2 - by 2 where a, b and c are positive
constants. z is the height of the surface above the xy plane. At point (1,1) what is the direction on the tangent plane at which the height of the surface is increasing most
rapidly?
(Ans: )
(4?)(2??222222b a b a k b a j b i a +++++?? ) 6. Practice
Edward & Penny: Problem sets 13.2, 13.7 and 13.8
Kreyzig: Problems Secs. 8.7 & 8.8
SJC/cc:10.1.02
tgT2:ms6
TE2002-EngrgMaths I
(2006/2007)
[SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE Department of Electrical and Computer Engineering
TE 2002 - ENGINEERING MATHEMATICS I
TUTORIAL 4
1. Evaluate the gradient of the following scalar fields. a and k are constant vectors. The vector k
z j y i x r ???++=.
(i)
r - a 2 (ii) r
1 (iii)
?
??a r 1 (iv) a . r
(v) log k x r
(Ans: (i) 2(r - a) , (ii) -r r -3 , (iii) -(r - a) r - a -3
(iv) a (v) -k x (k x r) k x r -2
2. Evaluate the divergence of the following vector fields. a is a constant vector.
(i) r r -3
(ii) a r - a n
(iii) 3r (a . r)r -5 - ar -3
3i) Prove that
(a) F . ?r = F
(b) ? ∫ f(u) du = f(u) ?u
ii) Evaluate the curl of (a) r , (b) ar -1 .
4. The vector H E , are functions of the position vector r and the time t defined by
]).([sin ct r n k e E ?=
ENGINEERING MATHEMATICS I
TUTORIAL 4
].([sin ct r n k h H o
o ?=με
where k, c are scalar constants, and n and h e , are constant unit vectors. Show that H and E satisfy Maxwell’s equation from empty space.
0.=?E 0.=?H t H E x o ???=?μ t
E H x o ??=?ε provided n and h e , are mutually perpendicular and right-handedly related. k is the propagation constant and c is the velocity of light. 0
1με=c
5. The potential, φ, at a point Q due to an electric dipole, of dipole moment p , as
shown in the diagram below is given by
3.41r
r p o επφ?= Find the electric field at Q. The electric field is related to potential by φ??=E
Practice
Edward & Penny: Problem set 15.1
Adams: Exercises 9.1 and 9.2
SJC/cc:10.1.02
tgT4:ms6
TE2002-EngrgMaths I
(2006/2007)
[SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE Department of Electrical and Computer Engineering
TE 2002 - ENGINEERING MATHEMATICS I
TUTORIAL 5
1. Evaluate the line integral
(i) ∫+?)
2,3,2()1,0,1(22dz y dy xz dx x , the straight line joining the two points.
(ii) ds yz x ∫)
2,0,0()0,1,1(2 , the curve x = cost, y = cost, z = √2 sint; 0 ≤ t ≤ π/2
where ds is the line segment.
(Ans: (i) 35
?=x (ii) ? )
2. Calculate the work done by F along curve C where F = 3xy j i ?2?? and C is the
piece of the hyperbola x 2 - y 2 = 1, z = 0 from (1,0,0) to (2,√3,0)
(Ans: √3)
3. Evaluate ∫c φ ds along curve C where φ = 4xy,
C : x = y = t, z = 2t ;
t : 1 → 2.
(Ans: 36
28 )
4. Sketch above the x-y plane, the surface Σ
(a) represented by the equation
Σ : x 2 + y 2 + z = 16
(b) a curve Γ lying on the surface Σ is given by the parametric equations
x = 1 + cosθ
y = sinθ
z = 14 - 2 cosθ
(i) Find the equation of the projection of the curve Γ on the x-y plane and hence
sketch the curve Γ.
(ii) Evaluate directly the line integral
∫Γ x dx + zdy - ydz
Γ for θ increasing from 0 to 2π.
over
curve
5. Find the angle between the surfaces at the given point of intersection P.
z = 3x2 + 2y2
z = -2x + 7y2
P is the point (1, 1, 5)
θ = 1.12 rad)
(Ans:
Practice:
Edward & Penny: Problem sets: 15.2 and 15.3
Adams: Exercises: 8.3 and 8.4
SJC/cc:10.1.02
tgT5:ms6
TE2002-EngrgMaths I
(2002/03) [SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE Department of Electrical and Computer Engineering
TE 2002 - ENGINEERING MATHEMATICS I
TUTORIAL 6
1.
Evaluate the following integrals
(i) ∫ ∫R (x 2 + y 2) dx dy , where R is the triangle with vertices (0,0), (1,0)
and (1,1).
(ii) ∫ ∫ ∫R u 2 v 2 w du dv dw , where R is the region u 2 + v 2 ≤ 1, 0 ≤ w ≤ 1.
(Ans: (i) 31, (ii) 48
π
2
(i) Evaluate the following integral
∫ ∫R (1 - x 2 - y 2) dx dy, where R is the region x 2 + y 2 ≤ 1,
with the aid of the subsitution x = r cos θ, y = r sin θ.
ii)
Find the Jacobian of the transformation
u = x -y ,
v = x + y .
Hence evaluate in the u - v coordinates the double integral
dy dx y x y x R +?∫∫cos
over the region R shown in the Fig. 1. Sketch the region S in the u - v plane
corresponding to region R. (Ans: (i) 2π
(ii) + ? sin1 )
y
x+y = 1
x
Fig. 1
3. Evaluate ∫ ∫B F . n ?dA where F = xy i ? - x 2j ? + (x + z) k
?. B is that portion of the plane 2x + 2y + z = 6 included in the first octant and n
? is a unit normal to B.
(Ans: 4
27 )
4. Find the flux of the vector field F = 3 i ?- 2y j ?
+ 6 k ? over the surface
S : x = 2u 2 - v ,
y = u - v 2 ,
z = u + v ,
and 0 ≤ u ≤ 1, 2 ≤ v ≤ 6 .
(Ans: 3
1960 )
5. The sphere x 2 + y 2 + z 2 = a 2 is pierced by the cylinder x 2 + y 2 = ay. Find the
volume enclosed by the two surfaces.
(Ans:
?322343πa ) Practice
Edward and Penny: Problem sets 15.6 & 15.7
SJC/cc:2.2.02
tgT6:ms6
TE2002-EngrgMaths I
(2002/03) [SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE Department of Electrical and Computer Engineering
TE 2002 - ENGINEERING MATHEMATICS I
TUTORIAL 7
1.
Find the volume of the region above the xy plane bounded by the paraboloid z = x 2 + y 2 and the cylinder x 2 + y 2 = a 2 . (Ans: 42a π
)
2. Use Gauss’s Theorem to evaluate ∫s F . n
?dS , where F = xy i ?+ j ? + (z + 1) k
? and S is the curved surface of the hemisphere x 2 + y 2 + z 2 = 9, z ≥ 0 .
n ? is the unit normal to the surface S. (Ans: 27π)
3. Evaluate ∫s ( ? x F ) ? n
?dS , where S is the open hemispherical surface x 2 + y 2 + z 2 = a 2 , z ≥ 0 ,
F = (1- ay) i ?+ 2y 2j ?
+ (x 2 + 1) k ? and n ?, the unit normal to dS, points away from the origin. Evaluate the integral.
(i) directly.
(ii) by means of Gauss’s Theorem and
(iii) by means of Stoke’s Theorem.
4. Determine whether or not F has a potential function. Where it exists, find a potential function for F and use it to evaluate ∫c F . d l for any smooth curve
‘C’ between the end points (1,1,1) to (3,1,4).
(i) F = e xyz [( 1 + yzx ) i ?+ x 2z j ?+ x 2y k
?]
(Ans : K xe xyz +=φ)
(ii) F = (y - 4xz) i ?+ x j ? + (3z 2 - 2x 2) k ? (Ans: -5 )
Practice Edward & Penny: Problem sets 15.6 and 15.7 SJC/cc:2.2.01 tgT7:ms6
1
TE2002-EngrgMaths I
(2004/05) [SJ CHUA]
NATIONAL UNIVERSITY Of SINGAPORE
Department of Electrical and Computer Engineering
ENGINEERING MATHEMATICS I
TUTORIAL 8
1. Examine for the extreme values for the following function:
x y x xy x V 72151532223+??+=
Ans: Max: 112; Min: 108
2. Show that a rectangular box of maximum volume with a prescribed surface area is a
cube. Express the volume V in term of the surface area, S. Ans: 6
6S S V = 3. The ground state energy of a particle in a pillbox (right-circular cylinder) is given by () +=222224048.22H R m h E π,
in which R is the radius and H is the height of the pillbox. The potential outside the
pillbox is infinite. Find the ratio of R to H that will minimize the energy for a fixed volume.
4. If y x , and z are subject to the constraints
11
222=++=++z y x z y x
` show that
333z y x ++ has a minimum value of 2717 and a maximum value of 1.
5. A closed rectangular box with a volume of 16 m 3 is made from two kinds of
materials. The top and bottom are made of materials costing 10 cents per m 2 and the sides from materials costing 5 cents per m 2. Find the dimension of the box so that the cost of materials is minimized.
SJC:2.2.02