(1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of
3d Integral Formulas Small Eu
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1. 11 1( ) ( )1
n nax b dx ax bn a
, 1n
2. ln | |dx xx
3. 1 ln | |dx ax bax b a
4. 1ax axae dx e , 1
lnx x
aa dx a
5. ln (ln 1)x dx x x
6. 1sin( ) cos( )aax b dx ax b
7. 1cos( ) sin( )aax b dx ax b
8. 2 cos sinsin2 2x x xx dx
9. 2 cos sincos2 2x x xx dx
10. 1 21 1sin sin cos sinn n nnx dx x x x dxn n
11. 1 21 1cos cos sin cosn n nnx dx x x x dxn n
12. tg ln | cos |x dx x
13. ctg ln | sin |dx x
14. 1 21tg tg tg1
n n nx dx x x dxn
15. 1 21ctg ctg ctg1
n n nx dx x x dxn
16. 1ln | tg |cos sin
dx xx x
17. 1ln | ctg |sin sindx x
x x
18. 2 2
1 1 tg 2 11 1cos cos cosn n n
x ndx dxn nx x x
19. 2 2
1 1 ctg 2 11 1sin sin sinn n n
x ndx dxn nx x x
20. 2arcsin arcsin 1x dx x x x
21. 2arccos arccos 1x dx x x x
22. 2arctg arctg ln 1x dx x x x
23. 2arcctg arcctg ln 1x dx x x x
24. 2 2 2 2 11 1( ) ( ) , 12 1
n nx x a dx x a nn
25. 2 22 2 lnxdx x a
x a
26. 2 2
1 arctgdx xa ax a
27. 122 2 ln | |a
dx x ax ax a
28. 2 2 2 2 313 ( )x x a dx x a
29. 2 2
2 2
xdx x ax a
30. 2 2 2 2 2 2 212 ln | |x a dx x x a a x x a
31. 2 2
2 2ln | |dx x x a
x a
32. 2 2
2 2
1 (ln ln( ))dx x x a aax x a
33. 2 2
2 2
xdx a xa x
34. 2 2 2 2 21 arcsin2
( )xa x dx x a x aa
35. 2 2
arcsindx xaa x
36. 21 ( 1)ax ax
axe dx e ax
37. 11n ax n ax n axna ax e dx x e x e dx
38. 2 21sin ( sin cos )ax ax
a be bx dx e a bx b bx
39. 2 21cos ( cos sin )ax ax
a be bx dx e a bx b bx
40. 1sinh( ) cosh( )aax dx ax
41. 1cosh( ) sinh( )aax dx ax
42. tanh( ) ln | cosh( ) |x dx x
Area with Double Integral: 2
12 1( ) ( )
x
A xA dxdy y x y x dx
(standard formula: y2 is top function, y1 is bottom function) Volume with Triple Integral: 2 1( , ) ( , )
xyV DV dxdydz z x y z x y dxdy (analogous)
Change of coordinates in integral
Polar 2D cossin
x ry r
[0, ], [0, 2 ]r a
dxdy rdrd
Cylindrical 3D cossin
x ry rz z
1 2[0, ], [0,2 ], [ , ]r a z z z dx dy dz r dr d dz
Spherical 3D cos sinsin sincos
xyz
[0, ], [0, 2 ], [0, ]r a 2 sindx dy dz d d d
Parameterize – Curves (x(t),y(t),z(t) and Surfaces (x(u,v),y(u,v),z(u,v))
Curves (1 parameter) Surfaces (2 parameters) Explicit function y=y(x)
( )x ty y t
[0, 2 ]t
Disk at z=b with radius R; center (p,q) ( , ) cos( )
[0, 2 ]( , ) sin( )
[0, ]( , )
x u v v u pu
y u v v u qv R
z u v b
Circle radius R; center (p,q) cos
sin
x t R t p
y t R t q
[0, 2 ]t
Cylinder with radius R and center (p,q)
1 2
( , ) cos( )[0, 2 ]
( , ) sin( )[ , ]
( , )
x t z R t pt
y t z R t qz z z
z t z z
Ellipse center (p,q); a and b cos
sin
x t a t p
y t b t q[0, 2 ]t
Sphere with radius R and center (p,q,s) ( , ) cos( )sin( )( , ) sin( )sin( )( , ) cos( )
x u v R u v py u v R u v qz u v R v s
[0,2 ][0, ]
uv
2 2( ) ( )az x p y q , z b (cut off paraboloid)
( , ) cos( )[0, 2 ]
( , ) sin( )[0, ]
( , )
x t z az t pt
y t z az t qz b
z t z z
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Line Integrals: Let C: (x(t),y(t),z(t)) parameterized curve with endpoints 1 2[ , ]t t
I. Arc length ds: 2
1
2 2 2( , , ) ( ( ), ( ), ( ))t
tC
f x y z ds f x t y t z t x y z dt
2D: 2
1
2 2( , ) ( ( ), ( ))t
tC
f x y ds f x t y t x y dt If y=y(x), parameterize with x=t and y=y(t).
II. *With dx, dy, dz : 2
1
( , , ) ( , , ) ( , , ) ( )t
tC
P x y z dx Q x y z dy R x y z dz P x Q y R z dt
**SPECIAL If rot[<P,Q,R>]=0 then find potential u: ( , )u P u Pdx f y zx
…
( , , ) ( , , ) ( , , ) ( ) ( )C
P x y z dx Q x y z dy R x y z dz u B u A ; A and B are endpoints of C.
2D: 2
1
( , ) ( , ) ( )t
tC
P x y dx Q x y dy P x Q y dt (set R=0 in all 3D formulas!)
Green’s theorem for closed curve C on x0y plane enclosing surface D.
( , ) ( , )DC
Q PP x y dx Q x y dy dxdyx y
(+ means D is on your “left” as you walk C)
Surface Integrals: I. S= (x(u,v),y(u,v),z(u,v)) ( , , ) ( ( , ), ( , ), ( , ))
uv
u vSD
f x y z dS f x u v y u v z u v dS dS du dv
where , ,uyx zdS u u u and , ,v
yx zdS v v v
S: z=z(x,y) 2 2( , , ) ( , , ( , )) ( ) ( ) 1xy
SD
z zf x y z dS f x y z x y dx dyx y
II. With dydz, dxdz, dxdy: ( , , ) ( , , ) ( , , ) , , ( )
uvu vS D
P x y z dydz Q x y z dxdz R x y z dxdy P Q R dS dS du dv
S: z=z(x,y) , , , ,1xyS D
z zPdydz Qdxdz Rdxdy P Q R dxdyx y
S: F(x,y,z)=c , , , ,1xy
FF yx
F FS Dz z
Pdydz Qdxdz Rdxdy P Q R dx dy
*** Stokes’ theorem for closed curve C in space enclosing surface S:
SC
dydz dxdz dxdy
Pdx Qdy Rdz x y zP Q R
(+ means S is on your left as you walk C with your head pointed in +z direction)
If we have 2D, then R=0 and Green’s thm.
****Gauss- Ostrogradsky’s theorem for closed surface S enclosing solid G
( , , ) ( , , ) ( , , )S G
P Q RP x y z dydz Q x y z dxdz R x y z dxdy dx dy dzx y z
Vector Calculus Scalar function: y=f(x), z=f(x,y), w=f(x,y,z) are scalar (real-valued functions) Vector field has components that are scalar functions.
1D: ( )x xF r x r i , 2D: ( , ), ( , )x y x yF r x y r x y r i r j ,
3D: ( , , ), ( , , ), ( , , )x y z x y zF r x y z r x y z r x y z r i r j r k
Note: ( , , )z zr r x y z is the z-component of the vector and it is a scalar function of the 3 independent variables x,y,z (Analogous for xr and yr )
Gradient, divergence and rotation are operators.
Gradient: scalar function vector field : nabla or del
f=f(x,y,z) grad( ) , ,f f f f f ff f i j kx y z x y z
Divergence: vector field scalar function
, ,x y zF r r r ( ) yx zrr rdiv F
x y z
We write: ( ) , , , ,x y zdiv F F r r rx y z
“scalar” product
Rotation (curl): vector field vector field
, ,x y zF r r r ( )
x y z
i j k
rot F Fx y z
r r r
“vector” product
2D: ,x yF r r use 0zr
Work: Let F be a vector field; C a curve parameterized by t, ( ( ), ( ), ( ))C x t y t z t , 1 2[ , ]t t t
*Then 2
1( ) ( ), ( ), ( )
t
C tW F ds F t x t y t z t dt (Analogous 1D:
b
a
W Fds )
Potential: Let F be a vector field with ( ) 0rot F . Then there exists f such that f F .
** ( ) ( )C C
W F ds f ds f B f A where A and B are the endpoints of C.
f is called the potential of F .
Circulation (Stokes): Let F and C a closed curve bounding the surface S. *** ( )
SC
Circ F ds rot F ndS , where n is the unit normal pointing “out” from C.
Divergence (Gauss) Theorem: Let F and closed surface S enclosing solid G. **** ( )
S GFlux FdS div F dxdydz