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EXERCISES CALCULUS II
Course ID: MI1124E
Chapter 1. Applications of Differential Calculus in Geometry
1.1. Curves
Exercise 1. Find an equation of the tangent line and normal line to the curves
2
a) y = e1−x at the intersection of this curve and the line y = 1
(
b)
x = 2t − cos(πt)
y = 2t + sin(πt)
at the point A corresponding to t = 1/2
2
2
c) x 3 + y 3 = 5 at the point M (8; 1)
Exercise 2. Evaluate the curvature at the arbitrary point of
(
2
2
2
b) x 3 + y 3 = a 3 (a > 0)
x = a(t − sin t)
a)
(a > 0).
y = a(1 − cos t)
c) r = aebϕ , (a, b > 0)
Exercise 3. Evaluate the curvature of the curve y = ln x at the point with positive abscissa x > 0. At
what point does the curve have maximum curvature? If x → ∞, how is the curvature?
Exercise 4. Find the envelope of the family of the following curves:
a) y =
x
c
+ c2
b) cx2 − 3y − c3 + 2 = 0
c) y = c2 (x − c)2
d) 4x sin c + y cos c = 1
1.2. Surfaces
Exercise 5. Suppose that p~(t), ~q(t), α(t) are differentiable functions. Prove that
a)
d
(~p(t)
dt
c)
d
(~p(t)~q(t))
dt
+ ~q(t)) =
d~
p(t)
dt
+
d~
q (t)
dt
= p~(t) d~qdt(t) +
d~
p(t)
~q(t)
dt
b)
d
(α(t)~p(t))
dt
d)
d
(~p(t)
dt
= α(t) d~pdt(t) + α0 (t)~p(t)
× ~q(t)) = p~(t) ×
d~
q (t)
dt
+
d~
p(t)
dt
× ~q(t)
Exercise 6. The curve C is given by ~r(t). Suppose that ~r(t) is a differentiable function and ~r0 (t) is always
perpendicular to ~r(t). Prove that C lies on the sphere with center the origin.
Exercise 7. Find an equation of the tangent line and normal plane of the curve

2

 x = a sin t
a)
y = b sin t cos t at the point corresponding to t = π4 , (a, b, c > 0)


z = c cos2 t

2

 x = 4 sin t
√
b)
at M (1; −2 3; 2)
y = 4 cos t


z = 2 sin t + 1
1
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Exercise 8. Evaluate the curvature of the curve


 x = cos t
at the point corresponding to t = π2
a)
y = sin t


z=t


 x = cos t + t sin t
b)
y = sin t − t cos t at the point corresponding to t = π


z=t
c) Evaluate the curvature at the point M (1; 0; −1) of the intersection curve of the cylinder 4x2 + y 2 = 4
and the plane x − 3z = 4
Exercise 9. Find an equation of the tangent plane and normal line of the surface
a) x2 − 4y 2 + 2z 2 = 6 at the point (2; 2; 3)
b) z = 2x2 + 4y 2 at (2; 1; 12)
c) ln(2x + y 2 ) + 3z 3 = 3 at the point (0; −1; 1)
d) x2 + 2y 3 − yz = 0 at (1; 1; 3)
Exercise 10. Find an equation of the tangent line and normal plane of the curve
(
(
x2 + y 2 = 10
2x2 + 3y 2 + z 2 = 47
a)
at
A(1;
3;
4)
b)
at B(−2; 1; 6)
y 2 + z 2 = 25
x2 + 2y 2 = z
Chapter 2. Multiple Integrals
2.1. Double Integrals
Exercise 11. Calculate the iterated integral by first reversing the order of integration
a)
R1
dx
−1
b)
R1
√
− 1−x2
c)
f (x, y)dy
√
2
R1−y
d)
dy
R2
1+y
R 2
dy
0
f (x, y)dx
e)
2−y
R2
f (x, y)dx
sin y
√
1+
0
R2
π
1−x
R 2
dy
0
Ry
f (x, y)dx +
0
R2
√
2
dy
√
2
R4−y
0
√
dx
0
R2x
√
f (x, y)dy
2x−x2
Exercise 12. Calculate the value of the multiple integral
RR y
a)
dxdy, D = {(x, y) ∈ R2 : 0 ≤ x ≤ 1; 0 ≤ y ≤ 2}
1+xy
D
b)
RR
c)
RR
d)
RR
x2 (y − x)dxdy, where D is a region bounded by two curves y = x2 and x = y 2
D
2xydxdy, where D is bounded by the curves x = y 2 , x = −1, y = 0 and y = 1
D
√
√
(x + y)dxdy, where D is bounded by x2 + y 2 ≤ 1, x + y ≥ 1
D
2
f (x, y)dx
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e)
RR
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|x + y|dxdy, where D = {(x, y) ∈ R2 : |x| ≤ 1; |y| ≤ 1}
D
RR
f)
(|x| + |y|)dxdy
|x|+|y|≤1
g)
R1
dx
1−x
R 2
0
0
xe3y
dy
1−y
Exercise 13. Find the limits of integration in the polar coordinates of
RR
f (x, y)dxdy, where D is a
D
domain
a) a2 ≤ x2 + y 2 ≤ b2
c)
x2
a2
+
y2
b2
b) x2 + y 2 ≥ 4x, x2 + y 2 ≤ 8x, y ≥ x, y ≤
√
3x
d) x2 + y 2 ≤ 2x, x2 + y 2 ≤ 2y
≤ 1, y ≥ 0, (a, b > 0)
Exercise 14. Evaluate the given integral by changing to polar coordinates
a)
RR
√
dx
0
b)
RR
c)
RR
d)
RR
e)
RR
RR2 −x2
ln(1 + x2 + y 2 )dy,
(R > 0)
0
xydxdy, where D is a half of surface: (x − 2)2 + y 2 ≤ 1, y ≥ 0
D
(sin y + 3x)dxdy, where D is a surface: (x − 2)2 + y 2 ≤ 1
D
|x + y|dxdy, where D is surface: x2 + y 2 ≤ 1
D
xy 2 dxdy, where D is a region bounded by the circles x2 + (y − 1)2 = 1 and x2 + y 2 − 4y = 0.
D
Exercise 15. Rewrite the following integral in terms of two variables u and v:
R1
0
dx
Rx
f (x, y)dy, if we let
−x
u = x + y, and v = x − y.
Exercise 16. Evaluate the given integrals
2xy + 1
RR
x2 y 2
2
2
p
dxdy, where D : x2 + y 2 ≤ 1 d)
+
≤1
|9x
−
4y
|dxdy,
where
D
:
4
9
1 + x2 + y 2
D
D
(
(
RR
y ≤ x2 + y 2 ≤ 2y
dxdy
RR
1 ≤ xy ≤ 9
√
b)
, where D :
(3x + 2xy)dxdy, where D :
e)
2
2
2
x ≤ y ≤ 3x
D (x + y )
y ≤ x ≤ 4y
D

2
2

 2x ≤ x + y√≤ 12
RR xy
c)
dxdy, where D :
x2 + y 2 ≥ 2 3y
2 + y2

x
D

x ≥ 0, y ≥ 0
a)
RR
2.2. Triple integrals
Exercise 17. Evaluate the triple integral


0 ≤ x ≤ 1
RRR
a)
zdxdydz, where the region V is bounded by
x ≤ y ≤ 2x
p

V

0 ≤ z ≤ 5 − x2 − y 2
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

1 ≤ y ≤ 2
RRR
2
b)
(3xy − 4xyz)dxdydz, where the region V is bounded by
0 ≤ xy ≤ 2

V

0≤z≤2


0 ≤ x ≤ 1
RRR
2
c)
xyeyz dxdydz, where the region V is bounded by
0≤y≤1

V
 2
x ≤z≤1
(
RRR 2
x2 + y 2 + z 2 ≤ 1
d)
(x + y 2 )dxdydz, where the region V is bounded by
x2 + y 2 − z 2 ≤ 0
V
RRR p
Exercise 18.
z x2 + y 2 dxdydz, where
V
a) V is bounded by the cylinder x2 + y 2 = 2x and planes: y = 0, z = 0, z = a, (y ≥ 0, a > 0)
b) V is a half of the sphere x2 + y 2 + z 2 ≤ a2 , z ≥ 0, (a > 0)
2
2
2
c) V l is a half of the elipsoid x a+y
+ zb2 ≤ 1, z ≥ 0, (a, b > 0)
2
√
RRR
Exercise 19.
ydxdydz, where V is a region bounded by the cone: y = x2 + z 2 and the plane
V
y = h, (h > 0)
Exercise 20. Evaluate the triple integral
a)
RRR x2
dxdydz, where V :
a2
V
b)
RRR
x2
a2
+
y2
b2
+
z2
c2
≤ 1 (a, b, c > 0)
(
(x2 + y 2 + z 2 )dxdydz, where V :
V
c)
1 ≤ x2 + y 2 + z 2 ≤ 4
x2 + y 2 ≤ z 2
RRR p
x2 + y 2 dxdydz, where V is a region bounded by x2 + y 2 = z 2 , z = −1
V
(
x2 + y 2 ≤ 1
dxdydz
d)
,
where
V
:
|z| ≤ 1
[x2 + y 2 + (z − 2)2 ]2
V
RRR p
x2 + y 2 + z 2 dxdydz, where V is the region bounded by x2 + y 2 + z 2 ≤ z
e)
RRR
V
2.3. Applications of triple integrals
Exercise 21. Evaluate the area of the region D bounded by the curves
(
d) r ≥ 1, r ≤ √23 cos ϕ
y 2 = x, y 2 = 2x
a)
x2 = y, x2 = 2y
(
e) x2 + (αx − y)2 ≤ 4 is a constant ∀α ∈ R
y = 0, y 2 = 4ax
b)
x + y = 3a, y ≤ 0, (a > 0).


x + y ≥ 1
(
2
2
2x ≤ x + y ≤ 4x
f)
x + 2y ≤ 2

c)

0≤y≤x
y ≥ 0, 0 ≤ z ≤ 2 − x − y
4
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Exercise 22. Evaluate the area of the region D is bounded by the curves (a > 0)
a) (x2 + y 2 )2 = 2a2 xy
b) r = a(1 + cos ϕ)
Exercise 23. Evaluate the volume of the region is bounded by the surfaces
(
c) z = 1 + x2 + y 2 , surface x2 + 4y 2 = 4 and the
z = 4 − x2 − y 2
a)
plane Oxy.
2z = 2 + x2 + y 2
p
d) az = x2 + y 2 , z = x2 + y 2 , (a > 0).
b) |x − y| + |x + 3y| + |x + y + z| ≤ 1.
Exercise 24. Evaluate the area of a part of the sphere x2 + y 2 + z 2 = 4a2 lying inside the surface
x2 + y 2 − 2ay = 0, (a > 0).
Chapter 3. Integrals depending on a parameter
3.1. Definite Integrals depending on a parameter
Exercise 25. Evaluate
a) lim
1+y
R
y→0 y
dx
1+x2 +y 2
b) lim
R2
y→0 0
x2 cos xydx
Exercise 26. Evaluate
a) I(y) =
R1
0
b) J(y) =
R1
arctan xy dx
c) K =
R1
0
xb −xa
dx,
ln x
0 < a < b.
ln(x2 + y 2 )dx
0
3.2. Improper Integrals depending on a parameter
Exercise 27. Show that the integral
R∞
a) I(y) = sin(yx)dx is convergent if y = 0 and is divergent if y 6= 0.
1
b) I(y) =
R∞ cos αx
0
c) I(y) =
R1
x2 +1
x−y dx =
0
d) I(y) =
+∞
R
0
Exercise 28.
is uniformly convergent on R.
R∞
ty−2 dt is convergent if y < 1 and divergent if y ≥ 1.
1
e−yx sinx x is uniformly convergent on [0, +∞).
a) Evaluate I(y) =
+∞
R
ye−yx dx(y > 0)
0
b) Prove that I(y) converges to 1 uniformly on [y0 , +∞) for all y0 > 0.
c) Explain why I(y) is not uniformly convergent on (0; +∞).
Exercise 29. Prove that
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a)
R∞
2
e−x =
0
b)
R∞ sin x
x
0
c)
R∞
√
π
2
dx =
f)
π
2
g)
e)
R∞
0
R∞
cos(x2 )dx =
0
0
sin x −yx
e dx
x
sin yx
dx
x(1+x2 )
x2
R∞ x sin yx
0
sin(x2 )dx =
+∞
R
R∞ 1−cos yx
0
0
d)
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=
π
2
√
2π
4
h)
R∞
a2 +x2
=
dx = π2 e−ay ,
2
e−yx dx =
0
i)
− arctan y
+∞
R |y|π
2
√
π
√ ,
2 y
a, y ≥ 0.
y>0
√
a
b
√
e− x2 − e− x2 dx = πb− πa, (a, b > 0).
0
= π2 (1 − e−y ),
y≥0
j)
+∞
R
arctan
x
−arctan xb
a
x
0
dx =
π
2
ln ab , (a, b > 0).
Exercise 30. Explain why
 +∞

Z
Z+∞
−yx
−yx
lim+ ye
dx.
lim+  ye dx 6=
y→0
y→0
0
0
Exercise 31. Evaluate (a, b, α, β > 0):
a)
+∞
R
0
b)
+∞
R
0
c)
+∞
R
0
d)
+∞
R
0
e)
+∞
R
0
f)
+∞
R
e−αx −e−βx
dx
x
2
e−αx −e−βx
x2
j)
+∞
R
2
dx
k)
+∞
R
−∞
i)
+∞
R
0
x
+∞
R
0
dx
(x2 +y)n+1
l)
Rπ
dx
2
e−ax −cos bx
dx
x2
ln(1 + y cos x)dx
0
sin bx−sin cx −ax
e dx
x
m)
cos bx−cos cx −ax
e dx
x
n)
e−ax cos(yx)dx
o)
+∞
R
2
e−x sin axdx
0
+∞
R
0
2
e−x cos(yx)dx
p)
sin xy
dx,
x
+∞
R R
0
0
h)
3 −e−bx3
e−ax
0
0
g)
+∞
R
+∞
R
+∞ −ax2
e
0
y≥0
cos bxdx
2
x2n e−x cos bxdx,
0
arctan(x+y)
dx
1+x2
q)
+∞
R
0
2 −e−bx2
e−ax
x
dx
r)
+∞
R
0
3.3. Euler Integrals
Exercise 32. Evaluate
6
sin ax cos bx
dx
x
sin ax sin bx
dx
x
n∈N
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π
a)
R2
sin x cos xdx
e)
√
x2n a2 − x2 dx, (a > 0)
f)
6
4
0
0
b)
Ra
+∞
R
2
x10 e−x dx
g)
R1
0
0
d)
+∞
R
0
0
c)
+∞
R
+∞
R
0
√
x
dx
(1+x2 )2
h)
1
dx
1+x3
xn+1
dx, (2
(1+xn )
1
√
dx, n
n
1−xn
+∞
R
0
< n ∈ N)
∈ N∗
x4
dx
(1+x3 )2
Exercise 33. On which intervals the following function
Z1
I(y) =
y 2 − x2
dx
(x2 + y 2 )2
0
is continuous.
Ry arctan x
dx.
y→1 0 x2 + y 2
Exercise 34. Find lim
Exercise 35. The integral I(y) =
R1 yf (x)
dx where f (x) is a possitive function, and continuous on
2
2
0 x +y
[0, 1].
π
Exercise 36. Given a function f (y) =
R2
ln (sin2 x + y 2 cos2 x)dx. Evaluate f 0 (1).
0
Exercise 37. Prove that the integral depending on the parameter
Z+∞
I(y) =
arctan(x + y)
dx
1 + x2
−∞
is a continuous and is differentiable function with respect to y. Evaluate I 0 (y) and then find the formula
for I(y).
Exercise 38. Evaluate the following integrals (with a, b, α, β > 0 and n ∈ Z+ ):
R1 xb − xa
dx
ln x
0
d)
R∞ e−αx − e−βx
b)
dx
x
0
e)
a)
c)
+∞
R
0
R1
xα (ln x)n dx
0
+∞
R
0
dx
(x2 + y)n+1
π
sin(bx) − sin(cx)
e−ax
dx
x
f)
R2
ln(1 + y sin2 x)dx, with y > −1
0
Exercise 39. Evaluate the following integrals
π
a)
R2
6
4
sin x cos xdx
b)
+∞
R (ln x)4
1
0
7
x2
dx
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c)
+∞
R
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2
x10 e−x dx
g)
−∞
0
d)
+∞
R
0
e)
√
x
dx
(1 + x2 )2
h)
+∞
R
0
f)
√
e2x 3 1 − e3x dx
R0
+∞
R
0
1
dx
1 + x3
Ra
√
x2n a2 − x2 dx, (a > 0, n ∈ N)
0
xn+1
dx, (2 < n ∈ N)
(1 + xn )2
i)
R1
0
1
√
dx, (2 ≤ n ∈ N)
n
1 − xn
Chapter 4. Line integrals
4.1. Line integral of the first kind
Exercise 40. Evaluate the following line integrals
√
R
a) (3x − y)ds, where C is a half of the circle y = 9 − x2
C
b)
R
(x − y)ds, where C is a circle x2 + y 2 = 2x
C
(
x = a(t − sin t)
(0 ≤ t ≤ 2π, a > 0)
y = a(1 − cos t)
C
(
Rp
x = a(cos t + t sin t)
(0 ≤ t ≤ 2π, a > 0)
x2 + y 2 ds, where C is a curve
d)
y = a(sin t − t cos t)
C
c)
R
y 2 ds, where C is a curve
4.2. Line integral of the second kind
Exercise 41. Evaluate the following line integrals
R 2
a)
(x − 2xy)dx + (2xy − y 2 )dy, where AB is a part of parabol y = x2 from A(1; 1) to B(2; 4).
AB
(
b)
R
(2x − y)dx + xdy, where C is a curve
C
x = a(t − sin t)
y = a(1 − cos t)
whose direction is increasing direction
of the parameter t, (0 ≤ t ≤ 2π, a > 0).
R
c)
2(x2 + y 2 )dx + x(4y + 3)dy, where ABCA is a broken line through the points A(0; 0), B(1; 1),
ABCA
d)
C(0; 2)
R
dx+dy
ABCDA
e)
|x|+|y|
4
R √
x2 +y 2 dx
C
2
, where ABCDA is a broken line through the points A(1; 0), B(0; 1), C(−1; 0), D(0; −1)
(
+ dy, where C is curve
√
x = t sin t
√
y = t cos t, (0 ≤ t ≤
π2
).
4
Exercise 42. Evaluate the following line integral
Z
(xy + x + y)dx + (xy + x − y)dy
C
in two ways: by computing it directly, and by Green’s formula, then compare the results, where C is a
curve:
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a) x2 + y 2 = R2
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b) x2 + y 2 = 2x
c)
x2
a2
+
y2
b2
= 1, (a, b > 0)
Exercise 43. Evaluate the following line integrals
I
y
x
a)
x2 (y + )dy − y 2 (x + )dx
4
4
x2 +y 2 =2x
I
b)
ex [(1 − cos y)dx − (y − sin y)dy], where OABO is a broken line through the points O(0; 0),
OABO
A(1; 1), B(0; 2)
I
c)
(xy + ex sin x + x + y)dx − (xy − e−y + x − sin y)dy
x2 +y 2 =2x
I
d)
x3
(xy 4 + x2 + y cos(xy))dx + ( + xy 2 − x + x cos(xy))dy, where C is a circle
3
(
x = a cos t
y = a sin t,
(a > 0)
C
Exercise 44. Using the line integral of the second kind in order to compute the area of the region bounded
by an arch of the cycloid: x = a(t − sin t), y = a(1 − cos t) and x-axis, (a > 0).
Exercise 45. Evaluate the following line integrals
(3;0)
Z
4
3
2 2
(2;2π)
Z
4
(x + 4xy )dx + (6x y − 5y )dy
a)
y
y y
y
y2
(1 − 2 cos )dx + (sin + cos )dy
x
x
x x
x
b)
(1;π)
(−2;−1)
Exercise 46. Evaluate the line integral
Z
I = (3x2 y 2 +
2
2
)dx + (3x3 y + 3
)dy
+1
y +4
4x2
L
where L is a curve y =
√
1 − x4 from A(1; 0) to B(−1; 0).
Exercise 47. Find the constant α such that the following integral is an independent of path in the domain
Z
(1 − y 2 )dx + (1 − x2 )dy
.
(1 + xy)α
AB
Exercise 48. Find the constants a, b such that (y 2 + axy + y sin(xy))dx + (x2 + bxy + x sin(xy))dy is total
differetial of the function u(x, y). Find the function u(x, y).
Exercise 49. Find the function h(x) such that the integral
Z
h(x)[(1 + xy)dx + (xy + x2 )dy]
AB
is independent of the path in the domain. With the function h(x) found, evaluate the integral above from
A(2; 0) to B(1; 2).
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SAMI - Hanoi University of Science and Technology
Contact: dotronghoang@gmail.com
Exercise 50. Find the function h(xy) in order to the integral
Z
h(xy)[(y + x3 y 2 )dx + (x + x2 y 3 )dy]
AB
is not dependent on the path in the domain. With the function h(xy) just found, find the above integral
from A(1; 1) to B(2; 3).
Chapter 5. Surface integrals
5.1. Surface integral of the first kind
Exercise 51. Evaluate the following surface integrals
a)
RR
b)
RR
c)
RR
(z + 2x +
S
x y z
4y
)dS, where S = {(x, y, z) : + + = 1, x ≥ 0, y ≥ 0, z ≥ 0}
3
2 3 4
(x2 + y 2 )dS, where S = {(x, y, z) : z = x2 + y 2 ; 0 ≤ z ≤ 1}
S
zdS, where S = {(x, y, z) : y = x + z 2 , 0 ≤ x ≤ 1, 0 ≤ z ≤ 1}
S
d)
RR
S
dS
, where S is a boundary of the tetrahedron x + y + z ≤ 2, x ≥ 0, y ≥ 0, z ≥ 0
(1 + x + y + z)2
5.2. Surface integral of the second kind
Exercise 52. Evaluate the following surface integrals
RR
a)
z(x2 + y 2 )dxdy, where S is a half of a sphere: x2 + y 2 + z 2 = 1, z ≥ 0, with the outward normal
S
vector.
y2
b)
ydzdx + z dxdy, where S is the sphere x +
+ z 2 = 1, x ≥ 0, y ≥ 0, z ≥ 0, with downward
4
S
orientation.
RR 2 2
c)
x y zdxdy, where S is the surface x2 + y 2 + z 2 = R2 , z ≤ 0 and has upward orientation.
RR
2
2
S
d)
RR
e)
RR
f)
RR
(y + z)dxdy, where S is the surface z = 4 − 4x2 − y 2 and z ≥ 0.
S
x3 dydz + y 3 dzdx + z 3 dxdy, where S is the sphere x2 + y 2 + z 2 = R2 and is oriented downward.
S
y 2 zdxdy + xzdydz + x2 ydzdx, where S is outside of the domain
S
(
g)
RR
x2 + y 2 ≤ 1, 0 ≤ z ≤ x2 + y 2
x ≥ 0, y ≥ 0
xdydz + ydzdx + zdxdy, where S is outside of the solid
S
(
(z − 1)2 ≥ x2 + y 2
a≤z≤1
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SAMI - Hanoi University of Science and Technology
Contact: dotronghoang@gmail.com
Exercise 53. Use Stokes Theorem to evaluate the line integral
Z
(x + y 2 )dx + (y + z 2 )dy + (z + x2 )dz,
C
where C is a boundary of the triangle (1; 0; 0), (0; 1; 0), (0; 0; 1), oriented counterclockwise when viewed
from above.
Exercise 54. Given the sphere S: x2 + y 2 + z 2 = 1 lies inside the cylinder
(
x2 + x + z 2 = 0
y ≥ 0,
RR
which is oriented outward. Prove that: (x − y)dxdy + (y − z)dydz + (z − x)dzdx = 0.
S
Chapter 6. Field theory
Exercise 55. Find the directional derivative of the function ~` of the fucntion u = x3 + 2y 3 + 3z 2 + 2xyz
~ B(3; 2; 3).
at A(2; 1; 1) with ~` = AB,
−−→
Exercise 56. Compute the length of vector gradu, with u = x3 + y 3 + z 3 − 3xyz at A(2; 1; 1). When is
−−→
−−→
gradu perpendicular to Oz, and when is gradu = 0?
−−→
Exercise 57. Evaulate gradu, with
p
1
u = r2 + + ln r where r = x2 + y 2 + z 2
r
Exercise 58. Find the directions in which the rate of change of the function
u = x sin z − y cos z
at the origin O(0, 0, 0) is maximum?
−−→
Exercise 59. Evaluate the angle of two vectors gradz at (3; 4) of the fucntions:
p
z = x2 + y 2
p
z = x − 3y + 3xy
Exercise 60. Which of the following fields are conservative and find their potential functions.
a) F~ = 5(x2 − 4xy)~i + (3x2 − 2y)~j + ~k
b) F~ = (yz − 3x2 )~i + xz~j + (xy + 2)~k
c) F~ = (x + y)~i + (x + z)~j + (z + y)~k
x~i + y~j + z~k
d) F~ = C p
, C 6= 0 is a constant
(x2 + y 2 + z 2 )3
x
2 ~
e) F~ = (arctan z + 4xyz)~i + (2x2 z − 3y 2 )~j + ( 1+z
2 + 2x y)k
Exercise 61. Let F~ = xz 2~i + yx2~j + zy 2~k. Find the flux of F across the surface S : x2 + y 2 + z 2 = 1, with
the outward direction.
Exercise 62. Let F~ = x(y + z)~i + y(z + x)~j + z(x + y)~k, L be an intersection of the cylinder x2 + y 2 + y = 0
and the half of sphere x2 + y 2 + z 2 = 2, z ≥ 0. Prove that the circulation of F~ around L equals to 0.
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