WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
2009
Ch11: Differentiability on R
n11.1: Partial Derivatives and Partial Integrals
Notation:
(1) E1× E2× . . . × En
:= {(x1,x2, . . . ,xn) :xj ∈ Ej, for j = 1, . . . , n}
(2) The partial derivative fxj exists at a pointa if and only if the limit
∂f
∂xj
(a) = lim
h→0
f (a + hej) −f (a) h
exists.
(3)
fxjxk := ∂2f
∂xk∂xj
:= ∂
∂xk
∂f
∂xj
.
WEN-CHINGLIEN Advanced Calculus (II)
Let V be a nonempty, open subset ofRn, let f : V →Rm, and let p ∈N.
(i) f is said to be Cp on V if and only if each partial derivative of f of order k ≤ p exists and is continuous on V .
(ii) f is said to be C∞on V if and only if f is Cp on V for all p ∈N.
Theorem (11.2)
Suppose that V is open inR2, that (a, b) ∈ V , and that f : V →R. If f is C1on V , and if one of the mixed second partial derivatives of f exists on V and is continuous at the point (a, b), then the other mixed second partial derivative exists at (a, b) and
∂2f
∂y ∂x(a, b) = ∂2f
∂x ∂y(a, b).
Note: These hypotheses are met if f ∈ C2(H).
WEN-CHINGLIEN Advanced Calculus (II)
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b), define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V .Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b), we have
(1) lim lim ∆(h, k )
= ∂2f
(a, b).
Proof.
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b),define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V . Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b), we have
(1) lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b), define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V .Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b), we have
(1) lim lim ∆(h, k )
= ∂2f
(a, b).
Proof.
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b), define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V . Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b), we have
(1) lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b), define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V . Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b),we have
(1) lim lim ∆(h, k )
= ∂2f
(a, b).
Proof.
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b), define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V . Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b), we have
(1) lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b), define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V . Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b),we have
(1) lim lim ∆(h, k )
= ∂2f
(a, b).
Proof.
Suppose that fyx exists on V and is continuous at the point (a, b). Consider ∆(h, k ) := f (a + h, b + k ) − f (a + h, b)
−f (a, b + k ) + f (a, b), define for |h|, |k | < r /√
2, where r > 0 is so small that Br(a, b) ⊂ V . Apply the Mean Value Theorem twice to choose scalars s, t ∈ (0, 1) such that
∆(h, k ) = k∂f
∂y(a + h, b + tk ) − k ∂f
∂y(a, b + tk )
=hk ∂f2
∂x ∂y(a + sh, b + tk ).
Since this last mixed partial derivative is continuous at the point (a, b), we have
(1) lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
On the other hand, the Mean Value Theorem also implies that there is a scalar u ∈ (0, 1) such that
∆(h, k ) = f (a + h, b + k ) − f (a, b + k ) − f (a + h, b) + f (a, b)
=h∂f
∂x(a + uh, b + k ) − h∂f
∂x(a + uh, b).
Hence, it follows from (1) that
k →0lim lim
h→0
1 k
∂f
∂x(a + uh, b + k ) − ∂f
∂x(a + uh, b)
= lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
Proof.
On the other hand, the Mean Value Theorem also implies that there is a scalar u ∈ (0, 1) such that
∆(h, k ) = f (a + h, b + k ) − f (a, b + k ) − f (a + h, b) + f (a, b)
=h∂f
∂x(a + uh, b + k ) − h∂f
∂x(a + uh, b).
Hence, it follows from (1) that
k →0lim lim
h→0
1 k
∂f
∂x(a + uh, b + k ) − ∂f
∂x(a + uh, b)
= lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
On the other hand, the Mean Value Theorem also implies that there is a scalar u ∈ (0, 1) such that
∆(h, k ) = f (a + h, b + k ) − f (a, b + k ) − f (a + h, b) + f (a, b)
=h∂f
∂x(a + uh, b + k ) − h∂f
∂x(a + uh, b).
Hence, it follows from (1) that
k →0lim lim
h→0
1 k
∂f
∂x(a + uh, b + k ) − ∂f
∂x(a + uh, b)
= lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
Proof.
On the other hand, the Mean Value Theorem also implies that there is a scalar u ∈ (0, 1) such that
∆(h, k ) = f (a + h, b + k ) − f (a, b + k ) − f (a + h, b) + f (a, b)
=h∂f
∂x(a + uh, b + k ) − h∂f
∂x(a + uh, b).
Hence, it follows from (1) that
k →0lim lim
h→0
1 k
∂f
∂x(a + uh, b + k ) − ∂f
∂x(a + uh, b)
= lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
On the other hand, the Mean Value Theorem also implies that there is a scalar u ∈ (0, 1) such that
∆(h, k ) = f (a + h, b + k ) − f (a, b + k ) − f (a + h, b) + f (a, b)
=h∂f
∂x(a + uh, b + k ) − h∂f
∂x(a + uh, b).
Hence, it follows from (1) that
k →0lim lim
h→0
1 k
∂f
∂x(a + uh, b + k ) − ∂f
∂x(a + uh, b)
= lim
k →0lim
h→0
∆(h, k )
hk = ∂2f
∂x ∂y(a, b).
Proof.
Since fx is continuous on Br(a, b), we can let h = 0 in the first expression. We conclude by definition that
∂2f
∂y ∂x(a, b)= lim
k →0
1 k
∂f
∂x(a, b+k )−∂f
∂x(a, b)
= ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
Since fx is continuous on Br(a, b), we can let h = 0 in the first expression.We conclude by definition that
∂2f
∂y ∂x(a, b) = lim
k →0
1 k
∂f
∂x(a, b+k )−∂f
∂x(a, b)
= ∂2f
∂x ∂y(a, b).
Proof.
Since fx is continuous on Br(a, b), we can let h = 0 in the first expression. We conclude by definition that
∂2f
∂y ∂x(a, b)= lim
k →0
1 k
∂f
∂x(a, b+k )−∂f
∂x(a, b)
= ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
Since fx is continuous on Br(a, b), we can let h = 0 in the first expression. We conclude by definition that
∂2f
∂y ∂x(a, b) = lim
k →0
1 k
∂f
∂x(a, b+k )−∂f
∂x(a, b)
= ∂2f
∂x ∂y(a, b).
Proof.
Since fx is continuous on Br(a, b), we can let h = 0 in the first expression. We conclude by definition that
∂2f
∂y ∂x(a, b) = lim
k →0
1 k
∂f
∂x(a, b+k )−∂f
∂x(a, b)
= ∂2f
∂x ∂y(a, b).
WEN-CHINGLIEN Advanced Calculus (II)
Prove that
f (x , y ) = (
xy xx22−y+y22
(x , y ) 6= 0
0 (x , y ) = 0
Theorem (11.4)
Let H = [a, b] × [c, d ] be a rectangle and suppose that f : H →R is continuous. If
F (y ) = Z b
a
f (x , y )dx , then F is continuous on [c, d ]; i.e.,
y →y0lim
y ∈[c,d ]
Z b a
f (x , y )dx = Z b
a
y →y0lim
y ∈[c,d ]
f (x , y )dx
for all y0∈ [c, d].
WEN-CHINGLIEN Advanced Calculus (II)
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b].Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0.Since H is compact, f is uniformly continuous on H. Hence, choose δ > 0 such that k(x , y ) − (z, w )k < δ and (x , y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k, it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ. We conclude that F is continuous on [c, d ].
Proof.
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0. Since H is compact, f is uniformly continuous on H.Hence, choose δ > 0 such that k(x , y ) − (z, w )k < δ and (x , y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k, it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ. We conclude that F is continuous on [c, d ].
WEN-CHINGLIEN Advanced Calculus (II)
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0.Since H is compact, f is uniformly continuous on H. Hence, choose δ > 0 such thatk(x, y ) − (z, w )k < δ and (x, y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k, it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ. We conclude that F is continuous on [c, d ].
Proof.
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0. Since H is compact, f is uniformly continuous on H.Hence, choose δ > 0 such that k(x , y ) − (z, w )k < δ and (x , y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k, it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ. We conclude that F is continuous on [c, d ].
WEN-CHINGLIEN Advanced Calculus (II)
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0. Since H is compact, f is uniformly continuous on H. Hence, choose δ > 0 such thatk(x, y ) − (z, w )k < δ and (x, y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k,it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ. We conclude that F is continuous on [c, d ].
Proof.
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0. Since H is compact, f is uniformly continuous on H. Hence, choose δ > 0 such that k(x , y ) − (z, w )k < δ and (x , y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k, it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ.We conclude that F is continuous on [c, d ].
WEN-CHINGLIEN Advanced Calculus (II)
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0. Since H is compact, f is uniformly continuous on H. Hence, choose δ > 0 such that k(x , y ) − (z, w )k < δ and (x , y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k,it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ. We conclude that F is continuous on [c, d ].
Proof.
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0. Since H is compact, f is uniformly continuous on H. Hence, choose δ > 0 such that k(x , y ) − (z, w )k < δ and (x , y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k, it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ.We conclude that F is continuous on [c, d ].
WEN-CHINGLIEN Advanced Calculus (II)
For each y ∈ [c, d ], f (·, y ) is continuous on [a, b]. Hence, by Theorem 5.10, F (y ) exists for y ∈ [c, d ].
Fix y0 ∈ [c, d] and let ε > 0. Since H is compact, f is uniformly continuous on H. Hence, choose δ > 0 such that k(x , y ) − (z, w )k < δ and (x , y ), (z, w ) ∈ H imply
|f (x, y ) − f (z, w )| < ε b − a. Since |y − y0| = k(x, y ) − (x, y0)k, it follows that
|F (y ) − F (y0)| ≤ Z b
a
|f (x, y ) − f (x, y0)|dx < ε for all y ∈ [c, d ] that satisfy |y − y0| < δ. We conclude that F is continuous on [c, d ].
Theorem (11.5)
Let H = [a, b] × [c, d ] be a rectangle inR2and let f : H →R. Suppose that f (·, y ) is integrable on [a, b] for each y ∈ [c, d ] and that the partial derivative fy(x , ·) exists on [c, d ] for each x ∈ [a, b]. If the two-variable function fy(x , y ) is continuous on H, then
d dy
Z b a
f (x , y )dx = Z b
a
∂f
∂y(x , y )dx for all y ∈ [c, d ].
Note: These hypotheses are met if f ∈ C1(H).
WEN-CHINGLIEN Advanced Calculus (II)
Recall that ”fy(x , ·) exists on [c, d ]” means that fy(x , ·) exists on (c, d ), and
fy(x , c) := lim
h→0+
f (x , c + h) − f (x , c)
h ,
fy(x , d ) := lim
h→0−
f (x , d + h) − f (x , d ) h
both exist (see Definition 4.6). Hence, it suffices to show that
h→0+lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ [c, d ), and
Proof.
Recall that ”fy(x , ·) exists on [c, d ]” means that fy(x , ·) exists on (c, d ), and
fy(x , c) := lim
h→0+
f (x , c + h) − f (x , c)
h ,
fy(x , d ) := lim
h→0−
f (x , d + h) − f (x , d ) h
both exist (see Definition 4.6).Hence, it suffices to show that
h→0+lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ [c, d ), and
WEN-CHINGLIEN Advanced Calculus (II)
Recall that ”fy(x , ·) exists on [c, d ]” means that fy(x , ·) exists on (c, d ), and
fy(x , c) := lim
h→0+
f (x , c + h) − f (x , c)
h ,
fy(x , d ) := lim
h→0−
f (x , d + h) − f (x , d ) h
both exist (see Definition 4.6). Hence, it suffices to show that
h→0+lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ [c, d ), and
Proof.
h→0−lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ (c, d ].The arguments are similar; we provide the details only for the first identity.
Fix x ∈ [a, b] and y ∈ [c, d ), and let h > 0 be so small that y + h ∈ [c, d ).Let ε > 0. By uniform continuity, choose a δ >0 so small that |y − c| < δ and x ∈ [a, b] imply
|fy(x , y ) − fy(x , c)| < ε/(b − a). By the Mean Value Theorem, choose a point c(x ; h) between y and y + h such that
F (x , y , h) := f (x , y + h) − f (x , y )
h = ∂f
∂y(x , c(x ; h)).
WEN-CHINGLIEN Advanced Calculus (II)
h→0−lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ (c, d ]. The arguments are similar; we provide the details only for the first identity.
Fix x ∈ [a, b] and y ∈ [c, d ), and let h > 0 be so small that y + h ∈ [c, d ). Let ε > 0. By uniform continuity, choose a δ >0 so small that |y − c| < δ and x ∈ [a, b] imply
|fy(x , y ) − fy(x , c)| < ε/(b − a). By the Mean Value Theorem, choose a point c(x ; h) between y and y + h such that
F (x , y , h) := f (x , y + h) − f (x , y )
h = ∂f
∂y(x , c(x ; h)).
Proof.
h→0−lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ (c, d ]. The arguments are similar; we provide the details only for the first identity.
Fix x ∈ [a, b] and y ∈ [c, d ), and let h > 0 be so small that y + h ∈ [c, d ).Let ε > 0. By uniform continuity, choose a δ >0 so small that |y − c| < δ and x ∈ [a, b] imply
|fy(x , y ) − fy(x , c)| < ε/(b − a).By the Mean Value Theorem, choose a point c(x ; h) between y and y + h such that
F (x , y , h) := f (x , y + h) − f (x , y )
h = ∂f
∂y(x , c(x ; h)).
WEN-CHINGLIEN Advanced Calculus (II)
h→0−lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ (c, d ]. The arguments are similar; we provide the details only for the first identity.
Fix x ∈ [a, b] and y ∈ [c, d ), and let h > 0 be so small that y + h ∈ [c, d ). Let ε > 0. By uniform continuity, choose a δ >0 so small that |y − c| < δ and x ∈ [a, b] imply
|fy(x , y ) − fy(x , c)| < ε/(b − a). By the Mean Value Theorem, choose a point c(x ; h) between y and y + h such that
F (x , y , h) := f (x , y + h) − f (x , y )
h = ∂f
∂y(x , c(x ; h)).
Proof.
h→0−lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ (c, d ]. The arguments are similar; we provide the details only for the first identity.
Fix x ∈ [a, b] and y ∈ [c, d ), and let h > 0 be so small that y + h ∈ [c, d ). Let ε > 0. By uniform continuity, choose a δ >0 so small that |y − c| < δ and x ∈ [a, b] imply
|fy(x , y ) − fy(x , c)| < ε/(b − a).By the Mean Value Theorem, choose a point c(x ; h) between y and y + h such that
F (x , y , h) := f (x , y + h) − f (x , y )
h = ∂f
∂y(x , c(x ; h)).
WEN-CHINGLIEN Advanced Calculus (II)
h→0−lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ (c, d ]. The arguments are similar; we provide the details only for the first identity.
Fix x ∈ [a, b] and y ∈ [c, d ), and let h > 0 be so small that y + h ∈ [c, d ). Let ε > 0. By uniform continuity, choose a δ >0 so small that |y − c| < δ and x ∈ [a, b] imply
|fy(x , y ) − fy(x , c)| < ε/(b − a). By the Mean Value Theorem, choose a point c(x ; h) between y and y + h such that
F (x , y , h) := f (x , y + h) − f (x , y )
h = ∂f
∂y(x , c(x ; h)).
Proof.
h→0−lim Z b
a
f (x , y + h) − f (x , y )
h dx =
Z b a
∂f
∂y(x , y )dx for y ∈ (c, d ]. The arguments are similar; we provide the details only for the first identity.
Fix x ∈ [a, b] and y ∈ [c, d ), and let h > 0 be so small that y + h ∈ [c, d ). Let ε > 0. By uniform continuity, choose a δ >0 so small that |y − c| < δ and x ∈ [a, b] imply
|fy(x , y ) − fy(x , c)| < ε/(b − a). By the Mean Value Theorem, choose a point c(x ; h) between y and y + h such that
F (x , y , h) := f (x , y + h) − f (x , y )
h = ∂f
∂y(x , c(x ; h)).
WEN-CHINGLIEN Advanced Calculus (II)
Since |c(x ; h) − y | = c(x ; h) − y ≤ h,it follows that if 0 < h < δ, then
F (x , y , h) − Z b
a
∂f
∂y(x , y )dx
≤ Z b
a
∂f
∂y(x , c(x ; h)) − ∂f
∂y(x , y )
dx < ε.
Therefore, d dy
Z b a
f (x , y )dx = Z b
a
∂f
∂y(x , y )dx .
Proof.
Since |c(x ; h) − y | = c(x ; h) − y ≤ h, it follows that if 0 < h < δ, then
F (x , y , h) − Z b
a
∂f
∂y(x , y )dx
≤ Z b
a
∂f
∂y(x , c(x ; h)) − ∂f
∂y(x , y )
dx < ε.
Therefore, d dy
Z b a
f (x , y )dx = Z b
a
∂f
∂y(x , y )dx .
WEN-CHINGLIEN Advanced Calculus (II)
Since |c(x ; h) − y | = c(x ; h) − y ≤ h, it follows that if 0 < h < δ, then
F (x , y , h) − Z b
a
∂f
∂y(x , y )dx
≤ Z b
a
∂f
∂y(x , c(x ; h)) − ∂f
∂y(x , y )
dx < ε.
Therefore, d dy
Z b a
f (x , y )dx = Z b
a
∂f
∂y(x , y )dx .
Proof.
Since |c(x ; h) − y | = c(x ; h) − y ≤ h, it follows that if 0 < h < δ, then
F (x , y , h) − Z b
a
∂f
∂y(x , y )dx
≤ Z b
a
∂f
∂y(x , c(x ; h)) − ∂f
∂y(x , y )
dx < ε.
Therefore, d dy
Z b a
f (x , y )dx = Z b
a
∂f
∂y(x , y )dx .
WEN-CHINGLIEN Advanced Calculus (II)
Since |c(x ; h) − y | = c(x ; h) − y ≤ h, it follows that if 0 < h < δ, then
F (x , y , h) − Z b
a
∂f
∂y(x , y )dx
≤ Z b
a
∂f
∂y(x , c(x ; h)) − ∂f
∂y(x , y )
dx < ε.
Therefore, d dy
Z b a
f (x , y )dx = Z b
a
∂f
∂y(x , y )dx .
Definition (11.6)
Let a < b be extended real numbers, let I be an interval in R, and suppose that f : (a, b) × I → R. The improper integral
Z b a
f (x , y )dx
is said to converge uniformly on I if and only if f (·, y ) is improperly integrable on (a, b) for each y ∈ I and given ε >0 there exist real numbers A, B ∈ (a, b) such that
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
< ε for all a < α < A, B < β < b and all y ∈ I.
WEN-CHINGLIEN Advanced Calculus (II)
Suppose that a < b are extended real numbers, that I is an interval inR, that f : (a, b) × I → R, and that f (·, y ) is locally integrable on the interval (a, b) for each y ∈ I. If there is a function M : (a, b) →R, absolutely integrable on (a, b), such that
|f (x, y )| ≤ M(x) for all x ∈ (a, b) and y ∈ I, then
Z b a
f (x , y )dx converges uniformly on I.
Proof.
Let ε > 0.By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I.Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I, we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
a
M(x )dx + Z b
B
M(x )dx < ε
WEN-CHINGLIEN Advanced Calculus (II)
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I. Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I, we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
M(x )dx + Z b
M(x )dx < ε
Proof.
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I.Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I, we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
a
M(x )dx + Z b
B
M(x )dx < ε
WEN-CHINGLIEN Advanced Calculus (II)
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I. Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I,we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
M(x )dx + Z b
M(x )dx < ε
Proof.
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I. Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I, we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
a
M(x )dx + Z b
B
M(x )dx < ε
WEN-CHINGLIEN Advanced Calculus (II)
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I. Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I,we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
M(x )dx + Z b
M(x )dx < ε
Proof.
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I. Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I, we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
a
M(x )dx + Z b
B
M(x )dx < ε
WEN-CHINGLIEN Advanced Calculus (II)
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I. Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I, we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
M(x )dx + Z b
M(x )dx < ε
Proof.
Let ε > 0. By hypothesis and the Comparison Test for improper integrals,Rb
a f (x , y )dx exists and is finite for each y ∈ I. Moreover, since M(x ) is improperly integrable on (a, b), there exist real numbers A, B such that
a < A < B < b and
Z A α
M(x )dx + Z b
B
M(x )dx < ε.
Thus for each a < α < A < B < β < b and each y ∈ I, we have
Z b a
f (x , y )dx − Z β
α
f (x , y )dx
≤ Z α
a
|f (x, y )|dx+
Z b β
|f (x, y )|dx
≤ Z A
a
M(x )dx + Z b
B
M(x )dx < ε
WEN-CHINGLIEN Advanced Calculus (II)
Suppose that a < b are extended real numbers, that c < d are finite real numbers, and that
f : (a, b) × [c, d ] →R is continuous. If F (y ) =
Z b a
f (x , y )dx converges uniformly on [c, d ]; i.e.,
y →y0lim
y ∈[c,d ]
Z b a
f (x , y )dx = Z b
a
y →y0lim
y ∈[c,d ]
f (x , y )dx
for all y0∈ [c, d].
Proof.
Let ε > 0 and y0∈ [c, d].Choose real numbers A, B such that a < A < B < b and
F (y ) − Z B
A
f (x , y )dx
< ε 3
for all y ∈ [c, d ]. By Theorem 11.4, choose δ > 0 such that
Z B A
(f (x , y ) − f (x , y0))dx
< ε 3 for all y ∈ [c, d ] that satisfy |y − y0| < δ. Then
WEN-CHINGLIEN Advanced Calculus (II)
Let ε > 0 and y0∈ [c, d]. Choose real numbers A, B such that a < A < B < b and
F (y ) − Z B
A
f (x , y )dx
< ε 3
for all y ∈ [c, d ].By Theorem 11.4, choose δ > 0 such that
Z B A
(f (x , y ) − f (x , y0))dx
< ε 3 for all y ∈ [c, d ] that satisfy |y − y0| < δ. Then
Proof.
Let ε > 0 and y0∈ [c, d]. Choose real numbers A, B such that a < A < B < b and
F (y ) − Z B
A
f (x , y )dx
< ε 3
for all y ∈ [c, d ]. By Theorem 11.4, choose δ > 0 such that
Z B A
(f (x , y ) − f (x , y0))dx
< ε 3 for all y ∈ [c, d ] that satisfy |y − y0| < δ. Then
WEN-CHINGLIEN Advanced Calculus (II)
|F (y ) − F (y0)|≤
F (y ) − Z B
A
f (x , y )dx +
Z B A
(f (x , y ) − f (x , y0))dx +
F (y0) − Z B
A
f (x , y0)dx
< ε 3 + ε
3+ ε 3 = ε for all y ∈ [c, d ] that satisfy |y − y0| < δ.
Proof.
|F (y ) − F (y0)| ≤
F (y ) − Z B
A
f (x , y )dx +
Z B A
(f (x , y ) − f (x , y0))dx +
F (y0) − Z B
A
f (x , y0)dx
< ε 3 + ε
3+ ε 3 = ε for all y ∈ [c, d ] that satisfy |y − y0| < δ.
WEN-CHINGLIEN Advanced Calculus (II)
Suppose that a < b are extended real numbers, that c < d are finite real numbers, that f : (a, b) × [c, d ] →R is continuous, and that the improper integral
F (y ) = Z b
a
f (x , y )dx
exists for all y ∈ [c, d ]. If fy(x , y ) exists and is continuous on (a, b) × [c, d ] and if
φ(y ) = Z b
a
∂f
∂y(x , y )dx
converges uniformly on [c, d ], then F is differentiable on [c, d ], and F0(y ) = φ(y ); i.e.,
Theorem (11.9) d dy
Z b a
f (x , y )dx = Z b
a
∂f
∂y(x , y )dx for all y ∈ [c, d ].
WEN-CHINGLIEN Advanced Calculus (II)