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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_12” 2011/5/23 1:04 page 53 #53
Real-valued Functions of Two Real Variables 12-53
5. Let u(r, θ) = f (x, y), where x = r cos θ and y = r sin θ . Express
f
x
2
+
f
y
2
in terms of
u
r
and
u
∂θ
.
Solution: Using the chain rule for several variables, we have (using
x
2
+ y
2
= r
2
and tan θ =
y
x
)
f
x
=
u
r
r
x
+
u
∂θ
∂θ
x
=
u
r
x
r
y
x
2
+ y
2
u
∂θ
f
y
=
u
r
r
y
+
u
∂θ
∂θ
y
=
y
r
u
r
+
x
x
2
+ y
2
u
∂θ
f
x
2
=
x
2
r
2
u
r
2
+
y
2
(x
2
+ y
2
)
2
u
∂θ
2
2xy
r(x
2
+ y
2
)
u
r
u
∂θ
f
y
2
=
y
2
r
2
u
r
2
+
x
2
(x
2
+ y
2
)
2
u
∂θ
2
+
2xy
r(x
2
+ y
2
)
u
r
u
∂θ
.
Thus
f
x
2
+
f
y
2
=
u
r
2
+
1
r
2
u
∂θ
2
.
UNSOLVED EXERCISES
1. Find out the domain and range of the function f (x, y) =
x + y.
2. Find out the domain ...
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Publisher Resources

ISBN: 9781299447561Publisher Website