Consider the closed and bounded set BCR2= [the ry-plane] defined by B = [0, 1] × [0, 1] = {(r, y) € R² : 0 < r ≤ 1 and 0 ≤ y ≤l}, and the closed and bounded set H CR²= [the ur-plane] defined by X H 1 {(u, v) € R² : 0 < u < √2 and -min{v, √2-u} ≤ r ≤ min{x, √2-u) Draw a large picture of B and draw a large picture of H. Also consider the locally linear function T: R2 R2 given by (x, y) = T(u, v) := ( √2 (u — v), f(x,y) := 1 for all (u, v) E R². The function rotates each point (u, v) by /4 radians, counterclockwise, around the origin (0,0). It is clear that T: HB is one-to-one and onto, and T: OH OB is one-to-one and onto. Hence, T: HB is one-to-one and onto. You may assume these facts. Next we define the function f: BR by f(1, 1) := 0, and 1 11/214 √√2 . 1:= (2) Calculate the integral ry (1) The function f is continuous at every point (r,y) € B, except (1,1). calculate (u + v) for all (r, y) € B\{(1,1)} . fff(x,y) dA. (ry) EB 1:= Sf_f(x,y) dA= f 1=0 (z.y) EB directly, by expanding · (g(u, v), h(u, v)), =1 [ ( f(x,y) dy) d f(x, y) := 1 1-ry as an infinite sum of non-negative polynomial functions gn: dr 8 f(x,y) = Σ 9n(x, y), for every (x, y) € B\{(1,1)}"; n=0
Consider the closed and bounded set BCR2= [the ry-plane] defined by B = [0, 1] × [0, 1] = {(r, y) € R² : 0 < r ≤ 1 and 0 ≤ y ≤l}, and the closed and bounded set H CR²= [the ur-plane] defined by X H 1 {(u, v) € R² : 0 < u < √2 and -min{v, √2-u} ≤ r ≤ min{x, √2-u) Draw a large picture of B and draw a large picture of H. Also consider the locally linear function T: R2 R2 given by (x, y) = T(u, v) := ( √2 (u — v), f(x,y) := 1 for all (u, v) E R². The function rotates each point (u, v) by /4 radians, counterclockwise, around the origin (0,0). It is clear that T: HB is one-to-one and onto, and T: OH OB is one-to-one and onto. Hence, T: HB is one-to-one and onto. You may assume these facts. Next we define the function f: BR by f(1, 1) := 0, and 1 11/214 √√2 . 1:= (2) Calculate the integral ry (1) The function f is continuous at every point (r,y) € B, except (1,1). calculate (u + v) for all (r, y) € B\{(1,1)} . fff(x,y) dA. (ry) EB 1:= Sf_f(x,y) dA= f 1=0 (z.y) EB directly, by expanding · (g(u, v), h(u, v)), =1 [ ( f(x,y) dy) d f(x, y) := 1 1-ry as an infinite sum of non-negative polynomial functions gn: dr 8 f(x,y) = Σ 9n(x, y), for every (x, y) € B\{(1,1)}"; n=0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Consider the closed and bounded set BC R² = [the zy-plane] defined by
B = [0, 1] × [0, 1] = {(r, y) € R² : 0 ≤ x ≤ 1 and 0 ≤ y ≤l},
and the closed and bounded set HCR²= [the ur-plane] defined by
H =
{(u,v) € R² : 0 ≤ u ≤ √2 and −min{v, √2-u} ≤ v ≤ min{x, √2-1)}.
Draw a large picture of B and draw a large picture of H.
Also consider the locally linear function T: R2 →R² given by
(x, y) = T(u, v) := ( √ (u – v), √/21 (1 + r))
v)
√√2
for all (u, v) € R².
The function T rotates cach point (u, v) by π/4 radians, counterclockwise,
around the origin (0,0). It is clear that T: Hº →Bº is one-to-one and
onto, and T: OH OB is one-to-one and onto. Hence, T: HB is
one-to-one and onto. You may assume these facts.
Next we define the function f: BR by f(1, 1) := 0, and
for all (x, y) € B\{(1,1)} .
1
f(x, y) :=
1-xy'
(1) The function f is continuous at every point (x, y) € B, except (1,1).
calculate
1:= SS f(x,y) dA.
(ry) € B
(2) Calculate the integral
1:=ſf_f(x,y) dA=
(z.y) EB
directly, by expanding
•I=1
x=0
J:
=1
= (g(u, v), h(u, v)),
y=0
1
f(x, y):
1. - xy
as an infinite sum of non-negative polynomial functions gn:
IM
f(x,y) dy) a
f(x,y) = Σ9n(x, y), for every (x,y) € B\{(1, 1)}*;
n=0
dx
and then using the fact that: (the integral of the infinite sum of these functions
9n(1, y) over B] equals (the infinite sum of the integral of each gn(x, y) over
B].
(3) Using Parts (1) and (2) above, calculate exactly the infinite sum](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c49c29d-b734-4891-8a07-7c122a77a78d%2F29982222-7f8f-4ba1-b1b1-6fa205b7fed8%2Fsyuq7w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the closed and bounded set BC R² = [the zy-plane] defined by
B = [0, 1] × [0, 1] = {(r, y) € R² : 0 ≤ x ≤ 1 and 0 ≤ y ≤l},
and the closed and bounded set HCR²= [the ur-plane] defined by
H =
{(u,v) € R² : 0 ≤ u ≤ √2 and −min{v, √2-u} ≤ v ≤ min{x, √2-1)}.
Draw a large picture of B and draw a large picture of H.
Also consider the locally linear function T: R2 →R² given by
(x, y) = T(u, v) := ( √ (u – v), √/21 (1 + r))
v)
√√2
for all (u, v) € R².
The function T rotates cach point (u, v) by π/4 radians, counterclockwise,
around the origin (0,0). It is clear that T: Hº →Bº is one-to-one and
onto, and T: OH OB is one-to-one and onto. Hence, T: HB is
one-to-one and onto. You may assume these facts.
Next we define the function f: BR by f(1, 1) := 0, and
for all (x, y) € B\{(1,1)} .
1
f(x, y) :=
1-xy'
(1) The function f is continuous at every point (x, y) € B, except (1,1).
calculate
1:= SS f(x,y) dA.
(ry) € B
(2) Calculate the integral
1:=ſf_f(x,y) dA=
(z.y) EB
directly, by expanding
•I=1
x=0
J:
=1
= (g(u, v), h(u, v)),
y=0
1
f(x, y):
1. - xy
as an infinite sum of non-negative polynomial functions gn:
IM
f(x,y) dy) a
f(x,y) = Σ9n(x, y), for every (x,y) € B\{(1, 1)}*;
n=0
dx
and then using the fact that: (the integral of the infinite sum of these functions
9n(1, y) over B] equals (the infinite sum of the integral of each gn(x, y) over
B].
(3) Using Parts (1) and (2) above, calculate exactly the infinite sum
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