55. A function f is called homogeneous of degree n if it satisfies the equation CHAPTER 14 Partial Derivatives f(tx, ty)= tf(x, y) for all t, where n is a positive integer and f has continuous second-order partial derivatives. (a) Verify that f(x, y) = x²y + 2xy² + 5y³ is homogeneous of degree 3. (b) Show that if f is homogeneous of degree n, then a²f əx² x². af af ax ду X +y [Hint: Use the Chain Rule to differentiate f(tx, ty) with respect to t.] 56. If f is homogeneous of degree n, show that + 2xy = a²f Әх ду nf(x, y) 2 + y² a²f 2 dy² = = n(n − 1)f(x, y) 14.6 Directional Derivatives and the Gradi 57. If SO 58. S NE Z F EURO 59. E d
55. A function f is called homogeneous of degree n if it satisfies the equation CHAPTER 14 Partial Derivatives f(tx, ty)= tf(x, y) for all t, where n is a positive integer and f has continuous second-order partial derivatives. (a) Verify that f(x, y) = x²y + 2xy² + 5y³ is homogeneous of degree 3. (b) Show that if f is homogeneous of degree n, then a²f əx² x². af af ax ду X +y [Hint: Use the Chain Rule to differentiate f(tx, ty) with respect to t.] 56. If f is homogeneous of degree n, show that + 2xy = a²f Әх ду nf(x, y) 2 + y² a²f 2 dy² = = n(n − 1)f(x, y) 14.6 Directional Derivatives and the Gradi 57. If SO 58. S NE Z F EURO 59. E d
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
![55. A function f is called homogeneous of degree n if it satisfies
the equation
f(tx, ty) = tf(x, y)
for all t, where n is a positive integer and f has continuous
second-order partial derivatives.
(a) Verify that f(x, y) = x²y + 2xy² + 5y³ is homogeneous
of degree 3.
(b) Show that if f is homogeneous of degree n, then
-60
rancisco
+²
a²f
2
əx²
14 Partial Derivatives
X
56. If f is homogeneous of degree n, show that
[Hint: Use the Chain Rule to differentiate f(tx, ty) with
respect to t.]
50
+ 2xy
Reno
af
af
+ y
əx ду
60
a²f
ах ду
J2
=
+
nf(x, y)
Las
14.6 Directional Derivatives and the Gradient V
The weather map in Figure 1 sho
the states of California and Neva
isothermals, join locations with
tion such as Reno is the rate of ch
east from Reno; Ty is the rate of
want to know the rate of change
a²f
dy ²
-
= n(n − 1)f(x, y)
57. If
58. S
0
Z
F
50. E
da
is
OF](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e094232-bbb7-4647-adcf-f347cdaaa755%2F9a7ec5e2-f6ec-4994-9895-873b5606ce53%2F8y1gg2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:55. A function f is called homogeneous of degree n if it satisfies
the equation
f(tx, ty) = tf(x, y)
for all t, where n is a positive integer and f has continuous
second-order partial derivatives.
(a) Verify that f(x, y) = x²y + 2xy² + 5y³ is homogeneous
of degree 3.
(b) Show that if f is homogeneous of degree n, then
-60
rancisco
+²
a²f
2
əx²
14 Partial Derivatives
X
56. If f is homogeneous of degree n, show that
[Hint: Use the Chain Rule to differentiate f(tx, ty) with
respect to t.]
50
+ 2xy
Reno
af
af
+ y
əx ду
60
a²f
ах ду
J2
=
+
nf(x, y)
Las
14.6 Directional Derivatives and the Gradient V
The weather map in Figure 1 sho
the states of California and Neva
isothermals, join locations with
tion such as Reno is the rate of ch
east from Reno; Ty is the rate of
want to know the rate of change
a²f
dy ²
-
= n(n − 1)f(x, y)
57. If
58. S
0
Z
F
50. E
da
is
OF
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