a, b " R, a < b.
[a, b] Pn = {x0, x1, ..., xn},
a = x0 < x1 < ... < xn = b.
xk = xk - xk-1 k Pn, 1 d" k d"
n.
Pn ´(Pn) = max{ xk : 1 d" k d" n}.
k Pn, 1 d" k d" n
x" " [xk-1, xk].
k
f [a, b] Pn
[a, b].
f [a, b] Pn
xk 1 d" k d" n,
n
Sn(f, Pn) := f(x") xk.
k
k=1
 f
[a, b] Pn [a, b].
f [a, b]
b
f(x)dx := lim Sn(f, Pn),
´(Pn)0
a
Pn [a, b]
x", 1 d" k d" n.
k
a b a
f(x)dx := 0 f(x)dx := - f(x)dx.
a a b
[a, b]
[a, b].
[a, b] R[a, b].
[0, 1].
f I = [a, b],
[c, d] ‚" I.
f I, Õ
Õ ć% f I.
a = t0 < t1 < tn-1 < tn = b f
[ti, ti+1], i " {0, 1, ..., n - 1}, f [a, b].
[a, b]
 f [a, b],
[a, b].
f g [a, b],
f + g
b b b
(f(x) + g(x))dx = f(x)dx + g(x)dx;
a a a
c " R, c · f
b b
c · f(x)dx = c · f(x)dx.
a a
f g [a, b],
h ć% f h f([a, b]);
|f|
f · g
f [a, b]
g f
g [a, b]
b b
g(x)dx = f(x)dx.
a a
f [a, b] c " [a, b],
b c b
f(x)dx = f(x)dx + f(x)dx.
a a c
f g
[a, b],
 "x " [a, b] f(x) d" g(x),
b b
f(x)dx d" g(x)dx.
a a
f : [a, b] R [a, b],
b
f(x)dx = F (b) - F (a),
a
F f
F (b) - F (a) F (x)|b.
a
f [-a, a], a > 0,
f
a a
f(x)dx = 2 f(x)dx;
-a 0
f
a
f(x)dx = 0;
-a
f T
a+T T
f(x)dx = f(x)dx.
a 0
f g
b b
f(x)g (x)dx = f(x)g(x)|b - f (x)g(x)dx.
a
a a
-

g : [Ä…, ²]na[a, b] [Ä…, ²],
g(Ä…) = a, g(²) = b,
f : [a, b] R [a, b],
b ²
f(x)dx = f(g(t))g (t)dt.
a Ä…
f [a, b]
m, M " R,
"x"[a,b] m d" f(x) d" M,
b
m(b - a) d" f(x)dx d" M(b - a).
a
f [a, b].
f [a, b]
b
1
f := f(x)dx.
b - a
a
f : [a, b] R
[a, b],
"c"[a,b] f = f(c),
b
"c"[a,b] f(x)dx = (b - a)f(c).
a
AB y = f(x)
x " [a, b], f [a, b].
|P | AB
y = 0, x = a x = b,
b
|P | = f(x)dx.
a
 AB P
x = x(t), y = y(t), t " [Ä…, ²],
x = a t = Ä…, x = b t = ², x y y
[a, b], AB
|P |
²
|P | = y(t) · x (t)dt.
Ä…
P
P = {(x, y) " R2 : a d" x d" b '" f1(x) d" y d" f2(x)},
f1 f2 [a, b] f1(x) d" f2(x)
x " [a, b],
b
|P | = (f2(x) - f1(x)) dx.
a
AOB AB
OA OB
AB
r = g(¸), ¸ " [¸1, ¸2],
g [¸1, ¸2]. |P |
AOB OA OB
¸2
1
|P | = (g(¸))2 d¸.
2
¸1
AB
AOB
²
1
|P | = (x(t) · y (t) - x (t) · y(t)) dt.
2
Ä…
 l
x = x(t), y = y(t), t " [Ä…, ²],
l x y
[a, b], |l| l
²
|l| = (x (t))2 + (y (t))2dt.
Ä…
l y = f(x), x " [a, b],
f [a, b], |l|
b
|l| = 1 + (f (x))2dx.
a
l
r = g(¸), ¸ " [¸1, ¸2],
g [¸1, ¸2],
|l| l
¸2
|l| = g2(¸) + (g (¸))2d¸.
¸1
S(x), x " [a, b],
V Ox X S
[a, b]. V
b
|V | = S(x)dx.
a
D = {(x, y) " R2 : a d" x d" b '" 0 d" y d" f(x)},
 f [a, b],
V D 0x
b
|V | = Ä„ f2(x)dx.
a
AB y = f(x), x "
[a, b], f
[a, b]. S AB
Ox
b
|S| = 2Ä„ f(x) 1 + (f (x))2dx.
a
AB
x = x(t), y = y(t), t " [Ä…, ²],
x y y [Ä…, ²],
AB S
AB Ox
²
|S| = 2Ä„ y(t) (x (t))2 + (y (t))2dt.
Ä…
 f
f : [a, +") R.
²
f [a, +") lim f(x)dx
²+"
a
+"
f(x)dx.
a
f [a, +")
+" -",
+" -".
f : (-", b] R. f
(-", b]
b b
f(x)dx := lim f(x)dx.
Ä…-"
-" Ä…
f : R R. f
(-", +")
+" a +"
f(x)dx := f(x)dx + f(x)dx,
-" -" a
a
f (-", +")
(-", +") a.
+"
dx
, a > 0, p > 1
xp
a
+" p d" 1.
b
dx
, b < 0,
xp
-"
f g
"x"[a,") 0 d" f(x) d" g(x).
+" +"
g(x)dx f(x)dx
a a
+" +"
f(x)dx g(x)dx
a a
f g
(-", b].
 f g ( )
[a, ")
f(x)
lim = k, 0 < k < +".
x+"
g(x)
+" +"
f(x)dx g(x)dx
a a
f : (a, b] R
a
f (a, b]
b b
f(x)dx := lim f(x)dx.
Ä…a+ Ä…
a
(a, b]
+" -",
+" -".
f : [a, b) R
b
b ²
f(x)dx := lim f(x)dx.
²b- a
a
f : [a, c) *" (c, b] R
c. f
[a, b]
b c b
f(x)dx := f(x)dx + f(x)dx.
a a c
f [a, b]
 f : (a, b) R
a b,
f (a, b)
b d b
f(x)dx := f(x)dx + f(x)dx,
a a d
d (a, b),
d.
b
dx
, b > 0 p < 1
xp
0
+" p e" 1.
f (a, b] [a, b)
R3.
R3
A, B, (a1, a2, a3), (b1, b2, b3);
-

-

a b [a1, a2, a3], [b1, b2, b3].
R
[0, 0, 0]
-

0 Ń.
[1, 0, 0], [0, 1, 0], [0, 0, 1] Ox, Oy
Oz i, j, k.
[a1, a2, a3] [b1, b2, b3]
a1 = b1, a2 = b2 a3 = b3.
-

-

a = [a1, a2, a3] b = [b1, b2, b3]
Ä… " R.
-

-

a b
-

-

a + b := [a1 + b1, a2 + b2, a3 + b3];
-

a Ä…
-

Ä…a := [Ä…a1, Ä…a2, Ä…a3].
-

a = [a1, a2, a3].
-

a
-
|| := a2 + a2 + a2.
a
1 2 3
-

-

a = [a1, a2, a3] b = [b1, b2, b3].
-

-

a b
-

-

a ć% b := a1 · b1 + a2 · b2 + a3 · b3.
-

-

a b
- - -

-
- -
a ć% b = ||| b | cos "(, b ).
a a
-

-

a b
- -

- -

a ć% b = 0 Ô! a Ä„" b .
-

-

a = [a1, a2, a3] b = [b1, b2, b3].
-

-

a b
i j k
-

-

a × b := =
a1 a2 a3
b1 b2 b3
a2 a3 a1 a3 a1 a2
i -
j + k.
b2 b3 b1 b3 b1 b2
-

-

a b
- -

- -
;
a × b = - b × a
- - -

- - -

a × b Ä„" a a × b Ä„" b ;
- - -

- - -
| × b | = ||| b | sin "(, b );
a a a
-
-
- -
||.
a × b = Ń Ô! a b
-
- -
, ,
a b c " R3.
-
- -
,
a b c
-

- .
-
( × b ) ć% c
a
-

-

a b
a1 a2 a3
-

-
-
( × b ) ć% c = .
a
b1 b2 b3
c1 c2 c3
P0(x0, y0, z0) " R3
-

-

a = [a1, a2, a3] b = [b1, b2, b3]
Å„Å‚
ôÅ‚
x = x0 + a1t + b1s
ôÅ‚
òÅ‚
,
y = y0 + a2t + b2s
ôÅ‚
ôÅ‚
ół
z = z0 + a3t + b3s
-
-
||Ä„||
t, s " R, Ä„ P0 " Ä„ a b
Ä„.
 P0(x0, y0, z0) " R3
-

n = [A, B, C].
A(x - x0) + B(y - y0) + C(z - z0) = 0
-

Ä„ P0 " Ä„ n Ä„" Ä„
Ä„.
P0(x0, y0, z0) " R3
-

a = [a1, a2, a3]
Å„Å‚
ôÅ‚
x = x0 + a1t
ôÅ‚
òÅ‚
,
y = y0 + a2t
ôÅ‚
ôÅ‚
ół
z = z0 + a3t
-
||l
t " R, l P0 " l a
l.
P0(x0, y0, z0) " R3
-

a = [a1, a2, a3].
x - x0 y - y0 z - z0
= =
a1 a2 a3
-
||l
l P0 " l a
l.
Ä„1 : A1x + B1y + C1z +
D1 = 0 Ä„2 : A2x + B2y + C2z + D2 = 0
A1x + B1y + C1z + D1 = 0
,
A2x + B2y + C2z + D2 = 0
l l.
 X
d : X × X R ( ),
"x,y"X d(x, y) e" 0,
"x,y"X (d(x, y) = 0 Ô! x = y),
"x,y"X d(x, y) = d(y, x).
"x,y,z"X d(x, z) d" d(x, y) + d(y, z).
(X, d), d
X (X, d),
d x, y " X, d(x, y),
x y.
X = Rn
n
d(x, y) := (xi - yi)2, x = (x1, ..., xn), y = (y1, ..., yn).
i=1
 d
(X, d)
n d
n = 1, R
d(x, y) = |x - y| x, y " R
(X, d)
a " X r
a r ( )
K(a, r) := {x " X : d(x, a) < r}.
A Ä…" X.
a " X A,
"r"R K(a, r) Ä…" A.
A Ä…" X,
x0 " X U(x0)
(X, d) x0.
B Ä…" X, X \ B
A
(X, d). A
´(A) := sup{d(x, y) : x, y " A}.
A ´(A) < "
A
 A
(X, d)
A Int(A).
A (X, d)
A = Int(A).
(X, d)
pn " X n " N.
(pn) (X, d)
p0 " X, lim pn = p0,
n"
lim d(pn, p0) = 0.
n"
(pk) Rn
p0 " Rn pk = (xk, ..., xk), k = 1, 2, ..., p0 = (x0, ..., x0).
1 n 1 n
lim pk = p0 Ô! "i"{1,...,n} lim xk = x0.
i i
k" k"
(X, d)
A ‚" X.
p0 " X A,
(pn),
"n"N pn " A '" pn = p0 '" lim pn = p0.

n"
A A .
p " A \ A A.
A
(X, d).
A A := A *" A .
 A (X, d)
A = A.
A ‚" X.
A (X, d),
A
A.
A Rn
A = "

(X, d), A1 ‚" A A2 ‚" A
A1 *" A2 = A,
A1 )" A2 *" A1 )" A2 = ".

A ‚" Rn
A.
Rn
 f A ‚" Rn
R n f : A R.
f p = (x1, ..., xn) " A f(p) f(x1, ..., xn).
f
{(x, y, z) " R3 : (x, y) " Df '" z = f(x, y)},
Df f.
f h " R
{(x, y) " Df : f(x, y) = h}.
n f : A R,
A ‚" Rn p0 A.
g f p0,
"µ>0"´>0"p"A (d(p, p0) < ´ Ò! |f(p) - g| < µ)
lim f(p) = g.
pp0
g n
 g g f
p0.
n f : A R, A ‚" Rn
p0 A.
g f p0,
"(p )‚"A "n"N pn = p0 '" lim pn = p0 Ò! lim f(pn) = g .

n
n" n"
n
" p0
f : A R, A ‚" R2 p0 =
(x0, y0) A.
g1 = lim lim f(x, y) g2 = lim lim f(x, y) ,
xx0 yy0 yy0 xx0
f.
p0 = (x0, y0)
g1 g2.
f g1 g2
g1 g2
f p0
g1 g2,
n
 n f : A R, A ‚" R2
p0 = (x0, y0) A.
f p0, lim f(p) = f(p0).
pp0
f A,
f n
n (x1, ..., xn) f(x1, ..., xn)
p0 = (x0, ..., x0)
1 n
k " {1, ..., n} xk f(x0, ..., x0 , xk, x0 , ..., x0) xk
1 k-1 k+1 n
x0 ( f p0
k
).
f, g
p0 " Rn,
f
f + g, f - g, f · g ( g(x) = 0 x " X)

g
p0.
f, g1, g2, ..., gn
g1, g2, ..., gn p0,
f q0 = (g1(p0), ..., gn(p0)),
f(g1, ..., gn) p0.
f,
p0, f(p0) > 0 ( f(p0) <
0), S(p0) p0
"p"S(p ) f(p) > 0 "p"S(p ) f(p) < 0 .
0 0
 f
D ‚" Rn,
"p "D"p "D f(p1) = inf f(p) '" f(p2) = sup f(p) .
1 2
p"D
p"D
f
D ‚" Rn.
f(D) R.
f D ‚" Rn,
"z"R inf f(p) d" z d" sup f(p) Ò! "p "D z = f(p0) .
0
p"D
p"D
 f : U(p0) R, U(p0)
-

p0 = (x0, ..., x0) " Rn h = [h1, ..., hn]
1 n
Rn.
-

f p0 h
-

f(p0 + t h ) - f(p0)
-
f(p0) := lim ,
h
t0
t
-

p0 + t h = (x0 + th1, x0 + th2, ..., x0 + thn).
1 2 n
-

Õ(t) := f p0 + t h ,
-

p0 " Rn, h Rn,
Õ(t) - Õ(0)
Õ (0) = lim .
t0
t
f(p0) = Õ (0).
-
h
(f + g)(p0) = f(p0) + g(p0).
- - -
h h h
-

z = f(x, y) h R2.
l
(x0, y0, 0),
-

h Oz.
f(x0, y0) = tg Å‚,
-
h
Å‚ l Oxy.
-

h .
f : U(p0) R, U(p0)
-

p0 " Rn. h Rn, r
f- (p0) fr- (p0)

h h
fr(p0) = rf(p0).
- -
h h
f(- - (p0) = f- (p0) + f- (p0).


h1+h2) h1 h2
f : U(p0) R, U(p0)
- -

p0 " Rn. h1, h2 Rn.
f- p0, f- p0,

h1 h2
f(- - = f- + f-

h1+h2 h1 h2
)(p0) (p0) (p0).
 f : U(p0) R, U(p0)
-

p0 " Rn, h Rn  > 0
-

p0 p0 +  h U(p0).
-

h , ¸ " (0, 1)
-

f p0 +  h - f(p0)
-

-
= f p0 + ¸ h .
h

e1, ..., en
Rn.
f p0 " Rn ei
f p0 i (
i )
"f
fx (p0) (p0).
i
"xi
f : U(p0) R, U(p0) ‚" Rn,
p0, f
f
D ‚" Rn, f
f
f.
f : A R, A ‚" Rn.
f p0
"f "f
("f)p := (p0) , ..., (p0) .
0
"x1 "xn
 p0
"f "f
"f := , ..., ,
"x1 "xn
f.
"f
i = 1, ..., n
"xi
-

p0 " Rn, f- (p0) h

h
-

f(p0) = ("f)p ć% h .
-
0
h
p0 " Rn, f : U(p0) R,
- -

U(p0) ‚" Rn. h = [h1, ..., hn] p0 + h " U(p0).
fx (p0) i " {1, ..., n}, f
i
p0,
-

f(p0 + h ) - f(p0) - [fx (p0)h1 + ... + fx (p0)hn]
1 n
lim = 0,
-

-

h 0
h
- - -

h h , h 0
hi 0 i " {1, ..., n}.
f p0.
R2.
-

p0 = (x0, y0) " R2 (x, y) " U(x0), h = ["x, "y],
"x = x - x0, "y = y - y0, f p0
f(x0 + "x, y0 + "y) - f(x0, y0) - [fx(p0)"x + fy(p0)"y]
lim = 0
"x0, "y0
("x)2 + ("y)2
 f p0.
"f(p0) = f(x0 + "x, y0 + "y) - f(p0), p0 = (x0, y0),
f p0
"f(p0) H" fx(p0)"x + fy(p0)"y.
f p0 " Rn,
f n fx (p0) i "
i
{1, ..., n} p0 " Rn, f p0.
p0 " Rn,
"f
f : U(p0) R, U(p0) ‚" Rn i = 1, ..., n
"xi
p0.
f p0
"2f " "
(p0) = (p0), i, j = 1, ..., n.
"xj"xi "xj "xi
"2f "2f
i = j, .
"xj"xi "x2
i
"2f
(p0) fx xj(p0).
"xj"xi i
i = j, fx xj

i
f : U(p0) R
"2f
p0,
"xj"xi
"2f "2f
(p0) = (p0).
"xj"xi "xi"xj
 f : U R,
"f
U ‚" Rn, i = 1, ..., n, t xi(t),
"xi
t " (Ä…, ²), (Ä…, ²) i = 1, ..., n, (x1(t), ..., xn(t)) "
U t " (Ä…, ²), f(x1, ..., xn) (Ä…, ²),
n
d "f dxi
f x1(t0), ..., xn(t0) = x1(t0), ..., xn(t0) t0 , t0 " (Ä…, ²).
dt "xi dt
i=1
"f
f : U R, U ‚" Rn, i = 1, ..., n U
"xi
(t1, ..., tm) xi(t1, ..., tm)
D ‚" Rm (x1(t1, ..., tm), ..., xn(t1, ..., tm)) " U (t1, ..., tm) " D,
f(x1, ..., xn) t0 = (t1, ..., tm) " D
n
" (x1(t0), ..., xn(t0)) "f "xk
= x1(t0), ..., xn(t0) t0 , j - 1, ..., m.
"tj "xk "tj
k=1
f : D R, D ‚" Rn.
f p0 " D
"U(p )‚"D"p"U(p ) f(p) e" f(p0);
0 0
"S(p )‚"D"p"S(p ) f(p) > f(p0).
0 0
(U(p0) p0, S(p0) p0)
p0
f :
U R, U ‚" R2,
(x0, y0),
"f "f
(x0, y0), (x0, y0),
"x "y
"f "f
(x0, y0) = 0, (x0, y0) = 0.
"x "y
n
f : U R, U ‚" R2,
(x0, y0)
"f "f
(x0, y0) = 0, (x0, y0) = 0,
"x "y
"2f "2f
(x0, y0) (x0, y0)
"x2 "x"y
det > 0,
"2f "2f
(x0, y0) (x0, y0)
"y"x "y2
(x0, y0) f
"2f
(x0, y0) > 0
"x2
"2f
(x0, y0) < 0.
"x2
yi =
fi(x1, .., xn), i " {1, ..., n}, G ‚" Rn,
"f1 "f1
"x1 "xn
"fn "fn
"x1 "xn
D(y1,...,yn)
.
D(x1,...,xn)
 F
F (x, y) = 0
y = y(x) I,
F (x, y(x)) = 0
x I. x =
x(y), y " J.
F
(x0, y0)
F (x0, y0) = 0,
Fy(x0, y0) = 0,

x0 y =
y(x) F (x, y) = 0, y(x0) = y0.
Fx(x, y(x))
y (x) = -
Fy(x, y(x))
x x0.
 F
(x0, y0), y = y(x)
x0
Fxx(Fy)2 - 2FxyFxFy + Fyy(Fx)2
y = - .
(Fy)3
F
(x0, y0)
F (x0, y0) = 0,
Fy(x0, y0) = 0,

Fx(x0, y0) = 0,
Fxx(x0, y0) = 0,

y = y(x) F (x, y) = 0,
y(x0) = y0 x0
xx
-F (x0,y0) > 0,
Fy(x0,y0)
xx
-F (x0,y0) < 0.
Fy(x0,y0)
Fx(x0, y0) = 0 Fx(x0, y0) = 0
Fxx(x0, y0) = 0 (x0, y0)

F (x0, y0) = 0.
x = x(y).
 K ‚" R2,
x = x(t), y = y(t) t " [Ä…, ²],
x y [Ä…, ²]
t " (Ä…, ²) K.
(x(Ä…), y(Ä…)) = (x(²), y(²)), K
D ‚" R2
y = y(x) x " [a, b] x = x(y) y " [c, d],
f
D ‚" R2 Pn D
n Di |Di|, i = 1, 2, ..., n
Di, Dj i = j

D = D1 *" D2 *" ... *" Dn.
´n = max{´(Di) : i " {1, 2, ..., n}}, ´(Di) Di,
Pn. Di
(xi, yi)
n
Sn = f(xi, yi) · |Di|.
i=1
{Pn}n"N D
lim ´n = 0
n"
{Sn}n"N
f, f D
f(x, y)dxdy.
D
f, D
D.
f
D ‚" R2,
f
D ‚" R2,
f
D ‚" R2
y = y(x)
x = x(y), D.
 f
D ‚" R2, g f
y = y(x) x = x(y)
D, g D
f(x, y)dxdy = g(x, y)dxdy.
D D
f g
D ‚" R2,
k " R k · f D
k · f(x, y)dxdy = k f(x, y)dxdy.
D D
f + g D
(f(x, y) + g(x, y))dxdy = f(x, y)dxdy + g(x, y)dxdy.
D D D
D ‚" R2
D1 D2
f D
D1 D2,
f(x, y)dxdy = f(x, y)dxdy + f(x, y)dxdy.
D D1 D2
f g
D ‚" R2 f(x, y) d" g(x, y)
(x, y) " D,
f(x, y)dxdy d" g(x, y)dxdy.
D D
 f
D ‚" R2 m d" f(x, y) d" M (x, y) " D,
m · |D| d" f(x, y)dxdy d" M · |D|.
D
f D
1
f = f(x, y)dxdy.
|D|
D
f
D ‚" R2. (x0, y0) " D,
f = f(x0, y0).
f D ‚" R2 Ox,
D = {(x, y) " R2 : a d" x d" b '" g(x) d" y d" h(x)},
b h(x)
f(x, y)dxdy = f(x, y)dy dx.
a g(x)
D
f D ‚" R2 Oy,
D = {(x, y) " R2 : a d" y d" b '" k(y) d" x d" l(y)},
d l(y)
f(x, y)dxdy = f(x, y)dx dy.
c k(y)
D
D
Ox Oy,
D = {(x, y) " R2 : a d" x d" b '" c d" y d" d}
 f D,
b d d b
f(x, y)dxdy = f(x, y)dy dx = f(x, y)dx dy.
a c c a
D
" D
Ouv Oxy.
" D T : " D
(x, y) = T (u, v) = (Åš(u, v), ¨(u, v)), (u, v) " ".
" T
T (") := {(x, y) : x = Åš(u, v), y = ¨(u, v), (u, v) " "}.
T
Åš ¨ ";
"
D.
(x, y) = T (u, v), x = Åš(u, v), y = ¨(u, v)
"
D " R2,
Åš ¨
",
f D,
J T D.
f(x, y)dxdy = f(Åš(u, v), ¨(u, v))|J(u, v)|dudv.
D "
 (Õ, r),
Õ Ox
p, 0 d" Õ d" 2Ä„ ( -Ä„ d" Õ d" Ä„),
r p 0 d" r < ".
(Õ, r)
V
V = {(x, y, z) " R3 : f1(x, y) d" z d" f2(x, y) '" (x, y) " D}
f1 f2 D,
|V | V
(f2(x, y) - f1(x, y)) dxdy.
D
S = {(x, y, z) " R3 : z =
f(x, y) '" (x, y) " D}, f D ‚" R2,
D f
D, S
S
F : D R ( f
D ‚" R2),
|S| S
2
|S| = 1 + (fx(x, y))2 + fy(x, y) dxdy.
D
 V ‚" R3
Oxy,
V = {(x, y, z) " R3 : (x, y) " Dxy '" h(x, y) d" z d" g(x, y)},
Dxy Oxy, h g
Dxy,
h(x, y) < g(x, y) (x, y) " Int(Dxy).
V Oxy,
Dxy V
Oxz
Oyz.
f
V ‚" R3 Pn V
n Vi |Vi|,
i = 1, 2, ..., n
Vi, Vj i = j

V = V1 *" V2 *" ... *" Vn.
´n = max{´(Vi) : i " {1, 2, ..., n}}, ´(Vi) Vi,
Pn. Vi
(xi, yi, zi) (i = 1, ..., n)
n
Sn = f(xi, yi, zi) · |Vi|.
i=1
 {Pn}n"N V
lim ´n = 0
n"
{Sn}n"N
f, f V
f(x, y, z)dxdydz.
V
f, V
V.
f
V ‚" R3,
f
V ‚" R3,
V.
f g
V ‚" R3,
k " R k · f V
k · f(x, y, z)dxdyz = k f(x, y, z)dxdydz.
V V
f + g V
(f(x, y, z) + g(x, y, z))dxdydz = f(x, y, z)dxdydz
V V
+ g(x, y, z)dxdydz.
V
 V ‚" R3
V1 V2
f V
V1 V2,
f(x, y, z)dxdydz = f(x, y, z)dxdydz + f(x, y, z)dxdydz.
V V1 V2
f g
V ‚" R3 f(x, y, z) d" g(x, y, z)
(x, y, z) " V,
f(x, y, z)dxdydz d" g(x, y, z)dxdydz.
V V
f
V ‚" R3 m d" f(x, y, z) d" M (x, y, z) " V,
m · |V | d" f(x, y, z)dxdydz d" M · |V |.
V
V
f V ‚" R3
1
f = f(x, y, z)dxdydz,
|V |
V
|V | V.
f
V ‚" R3.
(x0, y0, z0) " V,
f = f(x0, y0, z0).
 f
V = {(x, y, z) " R3 : (x, y) " Dxy '" h(x, y) d" z d" g(x, y)},
Oxy, h g
Dxy ‚" R2,
ëÅ‚ öÅ‚
g(x,y)
ìÅ‚
f(x, y, z)dxdydz = f(x, y, z)dz÷Å‚ dxdy.
íÅ‚ Å‚Å‚
V Dxy
h(x,y)
f
V ‚" R3 Oxy
V = {(x, y, z) " R3 : a d" x d" b '" g1(x) d" y d" g2(x) '" h1(x, y) d" z d" h2(x, y)},
îÅ‚ ëÅ‚ öÅ‚ Å‚Å‚
g2(x) h2(x,y)
b
ïÅ‚ ìÅ‚
f(x, y, z)dxdydz = f(x, y, z)dz÷Å‚ dyśł dx.
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
V a
g1(x) h1(x,y)
f
V = {(x, y, z) " R3 : a d" x d" b '" c d" y d" d '" p d" z d" q},
îÅ‚ ëÅ‚ öÅ‚ Å‚Å‚
b d q
ðÅ‚ íÅ‚
f(x, y, z)dxdydz = f(x, y, z)dzÅ‚Å‚ dyûÅ‚ dx.
V a c p
( ).
T = (x, y, z) : U V, U, V ‚" R3,
x = x(u, v, w)
y = y(u, v, w)
z = z(u, v, w)
V, x, y, z
U. f V T
"x "x "x
"u "v "w
D(x, y, z)
"y "y "y
JT = = = 0 U,

"u "v "w
D(u, v, w)
"z "z "z
"u "v "w
f(x, y, z)dxdydz =
V
f(x(u, v, w), y(u, v, w), z(u, v, w))|JT |dudvdw.
U
p = (x, y, z)
R3 (Õ, r, h),
Õ p
Oxy Ox, 0 d" Õ < 2Ä„ ( -Ä„ < Õ d" Ä„),
r p Oxy
0 d" r < ",
h p Oxy +
z > 0 - z < 0, -" < h < +".
(Õ, r, h)
R3.
W
x = r cos Õ
y = r sin Õ
z = h.
W, (Õ, r, h) (x, y, z)
JW = r.
p = (x, y, z)
R3 (Õ, Ć, r),
 Õ p
Oxy Ox, 0 d" Õ < 2Ä„ ( -Ä„ < Õ d" Ä„),
Ć p
Ä„
Oxy, -Ą d" Ć d" ,
2 2
r p 0 d" r < ".
(Õ, Ć, r)
R3.
S (Õ, phi, r) (x, y, z)
x = r cos Õ cos Ć
y = r sin Õ cos Ć
z = r sin Ć.
S
JS = r2 cos Ć.
 K = {r(t) : t " I}
r : I R2 r : I R3 r(t) = (x(t), y(t))
x y r K.
K r I.
K r r(t) =
(x(t), y(t)) x y
K r
{t " I : r (t) = 0}
K = r([a, b]), a < b r(a) = r(b).
K = r([a, b]), a < b
r([a, b)) r((a, b])
x = x1 + (x2 - x1)t
K : t " [0, 1]
y = y1 + (y2 - y1)t
Å„Å‚
ôÅ‚
x = x1 + (x2 - x1)t
ôÅ‚
òÅ‚
K :
y = y1 + (y2 - y1)t t " [0, 1]
ôÅ‚
ôÅ‚
ół
z = z1 + (z2 - z1)t
x = x0 + R cos t
K : t " [0, 2Ä„]
y = y0 + R sin t
x = x0 + a cos t
K : t " [0, 2Ä„]
y = y0 + b sin t
L
x = x(t), y = y(t) t " [Ä…, ²].
f L Pn
[Ä…, ²] n [ti, ti+1]
t0 = Ä… tn = ².
Ln L Pn L
Ai = (x(ti), y(ti)).
´Pn := max{"ti : i " {1, 2, ..., n}},
"ti [ti-1, ti], Pn.
´Li Li.
[ti, ti+1] Äi,
Bi = (x(Äi), y(Äi)) = (xB , yB )
i i
n
Sn = f(xB , yB ) · ´Li.
i i
i=1
{Pn}n"N [Ä…, ²]
lim ´Pn = 0 Äi,
n"
{Sn}n"N f,
f L
n
f(x, y)dl = lim f(xB , yB ) · ´Li.
i i
´Pn0
i=1
L
 L L
L1, L2, ..., Ln L,
f L
fdl := fdl + fdl + ... + fdl.
L L1 L2 Ln
f g
L
(f + g)dl = fdl + gdl
L L L
(c · f)dl = c · fdl.
L L
L =
{(x(t), y(t)) : t " [a, b]} f L
b
f(x, y)dl = f(x(t), y(t)) (x (t))2 + (y (t))2dt.
a
L
L
|L| = dl.
L
S
L ‚" R2 Oz f(x, y) e" 0
(x, y).
S
|S| = f(x, y)dl.
L
 L Á
M = Á(x, y)dl.
L
L Á
MSx = yÁ(x, y)dl,
L
MSy = xÁ(x, y)dl.
L
L Á
MSy MSx
x = , y = .
M M
Ox, Oy, Oz L ‚" R3 Á
Ix = (y2 + z2)Á(x, y, z)dl,
L
Iy = (x2 + z2)Á(x, y, z)dl,
L
Iz = (x2 + y2)Á(x, y, z)dl.
L
x = x(t), y = y(t), t " [Ä…, ²]
A =
(x(Ä…), y(Ä…)), B = (x(²), y(²))
B = (x(²), y(²)),
A = (x(Ä…), y(Ä…))
AB = -BA.
L
 A B.
x = x(t), y = y(t) t " [Ä…, ²].
P Q L Un
[Ä…, ²] n
[ti, ti+1] t0 = Ä… tn = ².
Ln L Un L
Ai = (x(ti), y(ti)).
´Un := max{"ti : i " {1, 2, ..., n}},
"ti [ti-1, ti], Un.
"xi = x(ti) - x(ti-1) "yi = y(ti) - y(ti-1).
´Li Li.
[ti, ti+1] Äi,
Bi = (x(Äi), y(Äi)) = (xB , yB )
i i
n
Sn = [P (xB , yB )"xi + Q(xB , yB )"yi] .
i i i i
i=1
{Un}n"N [Ä…, ²]
lim ´Un = 0 Äi,
n"
{Sn}n"N Sn,
P Q L
n
P (x, y)dx + Q(x, y)dy = lim P (xB , yB ) · "xi + Q(xB , yB ) · "yi.
i i i i
´Un0
i=1
L
P Q
L, P Q
-L,
P (x, y)dx + Q(x, y)dy = - P (x, y)dx + Q(x, y)dy.
-L L
L L
L1, L2, ..., Ln
L, f L
P (x, y)dx + Q(x, y)dy :=
L
P (x, y)dx + Q(x, y)dy + ... + P (x, y)dx + Q(x, y)dy.
L1 Ln
P Q L,
P Q
x = x(t), y = y(t), t " [Ä…, ²]
P (x, y)dx + Q(x, y)dy =
L
²
= [P (x(t), y(t))(x (t)) + Q(x(t), y(t))(y (t))] dt.
Ä…
F (x, y) = [P (x, y), Q(x, y)]
L
W = P (x, y)dx + Q(x, y)dy.
L
P, Q Py
Q x D
D L,
P (x, y)dx + Q(x, y)dy = (Q x(x, y) - Py(x, y))dxdy.
L D
"
y - xdx + xdy, L
L
A(2, 3) B(0, 9).
 D
D
D
P, Q
D,
L1 L2
D,
P (x, y)dx + Q(x, y)dy, P (x, y)dx + Q(x, y)dy
L1 L2
P Q C1
D.
P, Q
D
P (x, y)dx + Q(x, y)dy
F P (x, y)dx + Q(x, y)dy = F (B) -
AB
F (A).
P Q C1 D.
P (x, y)dx+Q(x, y)dy
"P "Q
= (x, y) " D.
"y "x