mate diferential complex
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v=x
t
v(t) =dr
dt
f : R R f(x) f df/dx x
f
(x0) =df(x)
dx |x0 = limh0
f(x0+h) f(x0)
(x0+h) x0
f f f(x) = x2
2 f(2) = 22 f : R R
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x0 f x0
f(x0+h) x0 f(x0)
x0
h
h x0 xo +h x0
f(x) = x2
f(x0) = limh0
f(x0+h) f(x0)(x0+h) x0
= limh0
(x0+h)2 x20(x0+h) x0
=
= limh0
(x20+ 2x0h+h2 x20
(x0+h) x0= lim
h0
2x0h+h2
h = lim
h0(2x0+h) = 2x0
2x0+h h
2x0 x2
2x
f, g: R R
(f g) =f g
(fg) =f g +fg
(f
g) =
fg fg
g2
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c = 0
(cx) =c
(xn) =nxn1
n N
n R
(sin(x)) =cos(x)
cos(x) = sin(x)
tg(x)
= 1 +tg(x)
(f(g(x))) =f(g(x)) g(x)
f(x) = sin(x3) x3 f(x) = Cos(x3) (x3) = 3x2Cos(x3)
f(x) =x3 7x 2 3x2 7 3x2 7 = 0 x=
7/3
f
f(x) = x2 f(x) = 2x 2x x2
f(x)
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F(x) = f(x)dxF(x) = f(x)
ba
f(x)dx
a b
Ox a b
ba
f(x)dx= F(b) F(a)
F f
f(x) = 2x x2 F(x) =x2 f(x)
31
2xdx= F(3) F(1) = 32 12 = 8
f(x)
f(x)
n [xi, xi+1]
xi = a+i x
x= (b a)/n
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xi xi+1 f(xi) x
A(f,a,b) =
ni=1
A(f, xi, xi+1) =
ni=1
x f(xi)
x 0
A(f,a,b) =
ba
f(x)dx= F(b) F(a)
v(t) = 2 (0, tmax) d= v tmax
v(t) =dx(t)
dt
dx(t) = v(t)dt
x
d=
tmax
0
dx(t) =
tmax
0
v(t)dt=
tmax
0
2dt= 2
tmax
0
dt= 2(tmax 0)
dx
t tmax
v(t) = t2
d=
tmax
0
dx(t) =
tmax
0
v(t)dt=
tmax
0
t2dt= t3
3|tmax0 =
tmax3
3
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t2 t3/3
=m
V
l
(x) =m
l
0 Lmax (x) = 2
(x) =dm(x)
dx
dm(x) = (x)dx
dm(x) Lmax
m=
Lmax
0
dm(x) =
Lmax
0
(x)dx=
Lmax
0
2dx= 2(Lmax 0)
(x) = Sin[x]
m=
Lmax
0
dm(x) =
Lmax
0
(x)dx=
Lmax
0
Sin(x)dx= Cos(x)|Lmax0 =Cos[Lmax]Cos[0] =Cos[Lmax
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M =J
M J
(r) = 8
J=
Lmax0
(r)r2dr =
Lmax0
r2dr= r3
3|Lmax0 =Lmax
3/3
Lmax/2, Lmax/2
J=
Lmax/2
Lmax/2
(r)r2dr=
Lmax/2
Lmax/2
r2dr= r3
3|Lmax/2Lmax/2 =
=/3(Lmax3
8
(Lmax)3
8 ) =
Lmax3
12
(x,y,z) =dm(x,y,z)
dV
dV
dV = dxdydz
m=
xmaxxmin
ymaxymin
zmaxzmin
dm(x,y,z) =
xmaxxmin
ymaxymin
zmaxzmin
(x,y,z)dV
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(x,y,z) =constant= 0
Ox
J=
xmax
xmin
ymax
ymin
zmax
zmin
(x,y,z)x2dV =0
xmax
xmin
ymax
ymin
zmax
zmin
x2dxdydz
0 x2 y z
J=... = 0
xmax
xmin
x2
dx(
ymax
ymin
zmax
zmin
dydz)
xy
ymax
ymin
zmax
zmin
dydz = z|zmaxzmin y|ymaxymin = (zmax zmin)(ymax ymin) = S
S
J=0S
xmaxxmin
x2dx= 0Sx3
3 |xmaxxmin =0S(xmax xmin)(
xmax3 xmin3
3(xmax xmin))
=mxmax3 xmin3
3(xmax xmin)
S(xmaxxmin)
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