61. u = cos(x²y)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
61
Transcribed Image Text:d fy with
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10-12
X
30 b
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Notice that fxy = fyx in Example 7. This is not just a coincidence. It turns out that the
mixed partial derivatives fxy and fyx are equal for most functions that one meets in prac-
tice. The following theorem, which was discovered by the French mathematician Alexis
Clairaut (1713-1765), gives conditions under which we can assert that fxy = fyx. The
proof is given in Appendix F.
2210-1²-2
X
Clairaut's Theorem Suppose f is defined on a disk D that contains the point
(a, b). If the functions fxy and fyx are both continuous on D, then
fxy(a, b) = fyx(a, b)
Partial derivatives of order 3 or higher can also be defined. For instance,
a
a³f
ay (ayax) - ay ² ax
ду
ду
![and
er hand-
and
them.
dr
42. f(x, y) = y sin(xy); f(1, 1)
43. f(x, y, z) = In
1
44. f(x, y, z) = x; f(e, 1, 0)
y
2x + 3y
57. v = sin(s² - 1²)
1
55. z =
16012960 0907 yil
45-46 Use the definition of partial derivatives as limits (4) to find
f(x, y) and f(x, y).
45. f(x, y) = xy² – x³y
+
51-52 Find az/ax and az/ay.
51. (a) z = f(x) + g(y)
52. (a) z = f(x) g(y)
(c) z = f(x/y)
69. w =
2
/x² + y² + z²
47-50 Use implicit differentiation to find az/ax and az/ay.
2
47. x² + 2y² + 3z² = 1
49. e² = xyz
X
/x² + y² + z²
68. V = In (r + s² + 1³);
y + 2z
nel 16
53-58 Find all the second partial derivatives.
53. f(x, y) = xy - 2x³y²
a ³W
du² dv
-; (1, 2, 2)
J³w
əz ay əx
46. f(x, y)
a³V
ar as at
59-62 Verify that the conclusion of Clairaut's Theorem holds, that
is, Uxy = Uyx.
SVIJ 160
59. u = x²y³ - y4
61. u = cos(x²y)
63-70 Find the indicated partial derivative(s).
63. f(x, y) = x4y² – x³y; fxxx, fxyx
64. f(x, y) = sin(2x + 5y); fyxy
65. f(x, y, z) = exyz²; fxyz
66. g(r, s, t) = e' sin(st); grst
67. W = √√√u + v²;
X
x + y²
48. x² - y² + z² = 2z = 4
obil 107.08
262
50. yz + x ln y = z²
(b) z=f(x + y) w sf.oe
(b) z = f(xy)
XX
54. f(x, y) = ln(ax + by)
56. T = e-2r cos 0 brut
58. w = √√√1 + uv²
2
d³ w
дх2 ду
60. u = exy sin y
62. u = ln(x + 2y)
vilicoq s
Appa
70. u = xªy¹zc;
SECTION 14.3 Partial Derivatives
71. If f(x, y, z) = xy²z³ + arcsin(x√z), find fxzy.
[Hint: Which order of differentiation is easiest?]
Jou
əx əy² əz³
noir
OU
72. If g(x, y, z)=√1 + x₂ + √√1 - xy, find gxyz. [Hint: Use a
different order of differentiation for each term.]
DOY
X
73. Use the table of values of f(x, y) to estimate the values of
f.(3, 2), f.(3, 2.2), and fy(3, 2).
2.5
3.0
3.5
y
1.8
12.5
YA
al tiuosio Iapitolo loll
104. I
18.1
20.0
(b) fy
(e) fyy
2.0
10.2
P,
17.5
22.4
2.2
9.3
15.9
74. Level curves are shown for a function f. Determine whether
the following partial derivatives are positive or negative at the
dipoint P.
bre(a) fib-
(c) fxx 01150291
(d) fxy
spr
bna
26.1
925
11/
10 8 6 4 2
x
"X35 - 9 moim rouborq asiguo dde
worl2 AS
75. Verify that the function u = e-a²k²t sin kx is a solution of the
heat conduction equation u, =
a²uxx.
AVA
76. Determine whether each of the following functions is a
solution of Laplace's equation uxx + Uyy = 0.
(a) u = x² + y²
(b) u = x² - y²
(c) u = x³ + 3xy²
(e) u = sin x cosh y + cos x sinh y
(f) u = e cos y - e cos x
(d) u = ln √√x² + y²
-
77. Verify that the function u = 1/√√x² + y² + z² is a solution of
the three-dimensional Laplace equation Uxx + Uyy + Uzz
= 0.
=
78. Show that each of the following functions is a solution of the
wave equation u₁t = a²uxx.
(a) u sin(kx) sin(akt)
(c) u = (x - at) + (x + at)6
=
(b) u = t/(a²t² - x²)
(d) u = sin(x - at) + In(x + at)
79. If f and g are twice differentiable functions of a single vari-
able, show that the function
u(x, t) = f(x + at) + g(x - at)
is a solution of the wave equation given in Exercise 78.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89e107ee-bcda-4baf-97d6-3bdf5aeb3272%2F7632c775-3f72-40ac-b315-ee7d5d5d3414%2Fd2ho2gh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:and
er hand-
and
them.
dr
42. f(x, y) = y sin(xy); f(1, 1)
43. f(x, y, z) = In
1
44. f(x, y, z) = x; f(e, 1, 0)
y
2x + 3y
57. v = sin(s² - 1²)
1
55. z =
16012960 0907 yil
45-46 Use the definition of partial derivatives as limits (4) to find
f(x, y) and f(x, y).
45. f(x, y) = xy² – x³y
+
51-52 Find az/ax and az/ay.
51. (a) z = f(x) + g(y)
52. (a) z = f(x) g(y)
(c) z = f(x/y)
69. w =
2
/x² + y² + z²
47-50 Use implicit differentiation to find az/ax and az/ay.
2
47. x² + 2y² + 3z² = 1
49. e² = xyz
X
/x² + y² + z²
68. V = In (r + s² + 1³);
y + 2z
nel 16
53-58 Find all the second partial derivatives.
53. f(x, y) = xy - 2x³y²
a ³W
du² dv
-; (1, 2, 2)
J³w
əz ay əx
46. f(x, y)
a³V
ar as at
59-62 Verify that the conclusion of Clairaut's Theorem holds, that
is, Uxy = Uyx.
SVIJ 160
59. u = x²y³ - y4
61. u = cos(x²y)
63-70 Find the indicated partial derivative(s).
63. f(x, y) = x4y² – x³y; fxxx, fxyx
64. f(x, y) = sin(2x + 5y); fyxy
65. f(x, y, z) = exyz²; fxyz
66. g(r, s, t) = e' sin(st); grst
67. W = √√√u + v²;
X
x + y²
48. x² - y² + z² = 2z = 4
obil 107.08
262
50. yz + x ln y = z²
(b) z=f(x + y) w sf.oe
(b) z = f(xy)
XX
54. f(x, y) = ln(ax + by)
56. T = e-2r cos 0 brut
58. w = √√√1 + uv²
2
d³ w
дх2 ду
60. u = exy sin y
62. u = ln(x + 2y)
vilicoq s
Appa
70. u = xªy¹zc;
SECTION 14.3 Partial Derivatives
71. If f(x, y, z) = xy²z³ + arcsin(x√z), find fxzy.
[Hint: Which order of differentiation is easiest?]
Jou
əx əy² əz³
noir
OU
72. If g(x, y, z)=√1 + x₂ + √√1 - xy, find gxyz. [Hint: Use a
different order of differentiation for each term.]
DOY
X
73. Use the table of values of f(x, y) to estimate the values of
f.(3, 2), f.(3, 2.2), and fy(3, 2).
2.5
3.0
3.5
y
1.8
12.5
YA
al tiuosio Iapitolo loll
104. I
18.1
20.0
(b) fy
(e) fyy
2.0
10.2
P,
17.5
22.4
2.2
9.3
15.9
74. Level curves are shown for a function f. Determine whether
the following partial derivatives are positive or negative at the
dipoint P.
bre(a) fib-
(c) fxx 01150291
(d) fxy
spr
bna
26.1
925
11/
10 8 6 4 2
x
"X35 - 9 moim rouborq asiguo dde
worl2 AS
75. Verify that the function u = e-a²k²t sin kx is a solution of the
heat conduction equation u, =
a²uxx.
AVA
76. Determine whether each of the following functions is a
solution of Laplace's equation uxx + Uyy = 0.
(a) u = x² + y²
(b) u = x² - y²
(c) u = x³ + 3xy²
(e) u = sin x cosh y + cos x sinh y
(f) u = e cos y - e cos x
(d) u = ln √√x² + y²
-
77. Verify that the function u = 1/√√x² + y² + z² is a solution of
the three-dimensional Laplace equation Uxx + Uyy + Uzz
= 0.
=
78. Show that each of the following functions is a solution of the
wave equation u₁t = a²uxx.
(a) u sin(kx) sin(akt)
(c) u = (x - at) + (x + at)6
=
(b) u = t/(a²t² - x²)
(d) u = sin(x - at) + In(x + at)
79. If f and g are twice differentiable functions of a single vari-
able, show that the function
u(x, t) = f(x + at) + g(x - at)
is a solution of the wave equation given in Exercise 78.
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