Parallelogram
Triangle
Parallelogram
𝐴𝑟𝑒𝑎 = 𝑏ℎ = 𝑎𝑏 𝑠𝑖𝑛𝜃
Trapezoid
You should remember this…
a
𝜋 = 3.1415
𝑒 = 2.7182
2 = 1.414
3 = 1.732
h
𝜃
1 𝑟𝑎𝑑 = 1808 /𝜋 = 57. 38
b
(𝑥 ± 𝑦)? = 𝑥 ? ± 2𝑥𝑦 + 𝑦 ?
𝑥 ? − 𝑦 ? = (𝑥 + 𝑦)(𝑥 − 𝑦)
Triangle
1
𝐴𝑟𝑒𝑎 = 𝑏ℎ
2
h
𝑛! = 1 ∙ 2 ∙ 3 ∙ ∙∙∙ ∙ 𝑛
0! = 1
b
Circle
Trapezoid
1
𝐴𝑟𝑒𝑎 = 𝑎 + 𝑏 ℎ
2
a
Quadratic Equation
𝑎𝑥 ? + 𝑏𝑥 + 𝑐 = 0
Sector
h
𝑥E,?
Sphere
Circle
𝐴𝑟𝑒𝑎 = 𝜋𝑟 ?
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 2𝜋𝑟
b
Sector
Cylinder
E
𝐴𝑟𝑒𝑎 = 𝑟 ? 𝜃
?
𝑆 = 𝑟𝜃
𝜃 - in rad.
r
Sphere
4
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 O
3
Cone
𝐴𝑟𝑒𝑎 = 4𝜋𝑟 ?
r
Cone
1
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 ? ℎ
3
𝑆 = 𝜋𝑟 𝑟 ? + ℎ?
(without base)
−𝑏 ± 𝑏 ? − 4𝑎𝑐
=
2𝑎
S
r
𝜃
r
Cylinder
Pyramid
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 ? ℎ
𝑆 = 2𝜋𝑟ℎ
(without circles)
r
h
Pyramid
1
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐴ℎ
3
h
r
h
A
1
TRIGONOMETRIC FUNCTIONS
tan 𝛼 =
sin 𝛼
cos 𝛼
cot 𝛼 =
cos 𝛼
sin 𝛼
1
cos 𝛼
sec 𝛼 =
𝑦 = sin 𝑥
csc 𝛼 =
1
sin 𝛼
sin(−𝑥) = − sin 𝑥
cos(−𝑥 ) = cos 𝑥
x
sin 𝑥
cos 𝑥
o
30
1/2
√3/2
o
45
√2/2
√2/2
60
1/2
√3/2
𝑦 = cos 𝑥
tan(−𝑥 ) = − tan 𝑥
cot(−𝑥 ) = − cot 𝑥
𝑥
tan 𝑥 cot 𝑥
o
30
√3/3
√3
o
45
1
1
60
√3
√3/3
𝑦 = tan 𝑥
O
sin 𝑥 = 𝑥 −
𝑦 = cot 𝑥
Y
Z
𝑥
𝑥
𝑥
+ − +⋯
3! 5! 7!
cos 𝑥 = 1 −
𝑥? 𝑥\ 𝑥]
+ −
2! 4! 6!
sin α ± β = sin α cos β ± cos α sin β
sin 2𝛼 = 2 sin 𝛼 cos 𝛼
cos α ± β = cos α cos β ∓ sin α sin β
cos 2𝛼 = cos ? 𝛼 − sin? 𝛼
Sines
sin? α + cos ? α = 1
Law of Sines
𝑎
𝑏
𝑐
=
=
sin 𝛼 sin 𝛽 sin 𝛾
Law of Cosines
𝑐 ? = 𝑎? + 𝑏 ? − 2𝑎𝑏 cos 𝛾
b
𝛼
c
𝛽
𝛾
a
complex
Complex Numbers
𝑖 = −1
𝑥 + 𝑖𝑦 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) = 𝑟𝑒 de
𝑒 df = −1 (Euler’s Formula)
r
y
𝜃
x
De Moivre’s Theorem (m-real number)
𝑟 cos 𝜃 + 𝑖 sin 𝜃
g
= 𝑟 g cos 𝑚𝜃 + 𝑖 sin 𝑚𝜃
2
f (x)
f (x) =
f (x) = 5 x
f (x) = 2 x
3. The relationship between exponential functions and logarithm functions
EXPONENTIAL & LOGARITHMIC
We FUNCTIONS
can see the relationship between the exponential function f(x) = e
x
and the logarithm
function f(x) = ln x by looking at their graphs.
𝑥 j 𝑥 k = 𝑥 jlk
(𝑥 j )k = 𝑥 jk
(𝑥𝑦)j = 𝑥 j 𝑦 j
q
𝑥 = 𝑥 E/r
q
q
𝑥/𝑦 = 𝑥 / q 𝑦
mn
= 𝑥 jpk
𝑥 = 1/𝑥 j
q
𝑥 g = 𝑥 g/r
𝑥 s = 1 (𝑥 ≠ 0)
mo
pj
ln 𝑥𝑦 = ln 𝑥 + ln 𝑦
𝑥
ln = ln 𝑥 − ln 𝑦
𝑦
ln 𝑥 r = 𝑛 ln 𝑥
𝑒
x
f (x) = x
f (x)
ln x axis.
In general, f(x) = (1/a)x = a−x is a reflection of f(x) = ax inf (x)
the=f(x)
x
e
A particularly important example off (x)
an =exponential
function arises when a = e. You might recall
that the number e is approximately equal to 2.718. The function f(x) = ex is often called ‘the’
x
exponential function. Since e > 1 and 1/e < 1, we can sketch the graphs of the exponential
x
−x
x
functions f(x) = e and f(x) = e = (1/e) .
±de
= cos 𝜃 ± 𝑖 sin 𝜃
𝑒 de − 𝑒 pde
sin 𝜃 =
2𝑖
de
𝑒 + 𝑒 pde
cos 𝜃 =
2
f (x)
You can see straight away that the logarithm function is a reflection of the exponential function
in the line represented by f(x) = x. In other words, the axes have been swapped: x becomes
f(x), and f(x) becomes x.
x
−x
f (x) = e
f (x) = e
HYPERBOLIC
Key Point
FUNCTIONS
x f(x) = ln x.
The exponential function f(x) = ex is the inverse of the logarithm function
𝑥? 𝑥O
𝑒 =1+𝑥+ + +⋯
HYPERBOLIC
FUNCTIONS
2! 3! HYPERBOLIC
FUNCTIONS
GRAPHS OF HYPERBOkfC
m
29
FUNCltONS 29
Exercises
1. Sketch the graph of the function f(x) =4a
www.mathcentre.ac.uk
HYPERBOLIC FUNCTIONS
x
GRAPHS
pm
𝑒m − 𝑒
2
sinh x
sinh 𝑥 =
8.49
8.49
y =
y = sinh x
𝑒 +𝑒
8.50 2
axes.
tanh 𝑥 =(a)
cosh 𝑥 =
8.50
y = coshx
y = coshx
8.52
𝑦 = sinh 𝑥
1
77
10
8.53
-1
10
𝑦 -1
= tanh 𝑥
X
/i
8.51
X
7
1
10
0
sinh
8.54
𝑥±
𝑦 = sinh 𝑥 cosh 𝑦 ± cosh 𝑥 sinh 𝑦
y = csch x
cosh 𝑥 ± 𝑦 = cosh 𝑥 cosh 𝑦 ± sinh 𝑥 sinh 𝑦
0
sinh
y 𝑥
=
Fig. 8-5
tanh x
c mathcentre 2009
⃝
Fig. 8-2
8.53
Fig. 8-3
y = sech x
siny 𝑖𝑥 = 𝑖 sinh 𝑥 8.54
cos 𝑖𝑥 = cosh
𝑥
sech x
y = csch
8.54
csch x
tan 𝑖𝑥
= 𝑖 tanh 𝑥y = cot
𝑖𝑥— 𝑖 coth 𝑥
Y
sinh 𝑖𝑥 = 𝑖 sin 𝑥
cosh
𝑖𝑥 = cos 𝑥 Y
X tanh 𝑖𝑥 = 𝑖 tan 𝑥 coth 𝑖𝑥 = X
−𝑖 cot 𝑥 \
0
\
0 X
X
0
y = csch x
8.54
Fig. 8-3
Fig. 8-3
-1
Fig. 8-4
Fig. 8-4 Fig. 8-4Y
1
6
1
3
8.51
y = tanh x
𝑦 = coth 𝑥
Fig. 8-3
\
coth(d)𝑥a ==
(c) a =
Fig. 8-2
8.53
y = sech x y =
y
y
1
𝑦y== cosh
𝑥
coth x
Fig. 8-2
/i
1
sinh(−𝑥) = − sinh 𝑥
cosh −𝑥 = cosh 𝑥
www.mathcentre.ac.uk
9
tanh −𝑥 = − tanh
𝑥
coth −𝑥 = − coth 𝑥
8.52
y x= coth x
y = coth
/i
X
a = 3 (b) a = 6
cosh
𝑥
Fig. S-l
y = tanh x
Fig. S-l Fig. S-l
8.52
1
(e) a = 6
3
GRAPHS
HYPERBOkfC
8.51
8.49
y OF
= sinh
x
8.50FUNCltONS
y = coshx
y = tanh x
OF HYPERBOkfC
FUNCltONS
2.
Sketch
the
graph
of
the
function
f(x)
=
log
x
for
the following values of a, on the same
a
m
pm
sinh 𝑥
cosh 𝑥
FUNCltONS
8.51
for the following
of a, on 2009
the same axes.
c values
⃝
mathcentre
(a) a = 3 (b) a = 6 (c) a = 1 (d) a =
29
x
x
()
( 21 ) x
f (x) =
NCTIONS
C
1 x
5
L
X0
x
\
L
0
L
X
Fig. 8-6
Fig. 8-5
sinh 2𝑥 = 2 sinh 𝑥 cosh 𝑥
cosh 2𝑥 = cosh? 𝑥 + sinh? 𝑥
Fig.
Fig. 8-5
Fig. 8-6
cosh? 𝑥 −
sinh? 𝑥 =
1 FUNCTIONS
iNVERSE
HYPERROLIC
Y
8-6
L
If x = sinhHYPERROLIC
g, then y = sinh-1 xFUNCTIONS
is called the inverse hyperbolic sine of x. Similarly we
iNVERSE
iNVERSE
FUNCTIONS
X otherHYPERROLIC
The inverse hyperbolic functions are multiple-valued
and.
inverse hyperbolic functions.
3
case of inverse trigonometric functions [sec page 171 we restrict ourselves to principal values
they ean be considered as single-valued.
If
x
=
sinh
g,
then
y
=
sinh-1
x the
is called
inverse sine
hyperbolic
sine of we
x. define
Similarly
If x = sinh g, then y = sinh-1 x is called
inverse the
hyperbolic
of x. Similarly
the we define the
TheThe
following
list shows
the principal
values [unless
otherwise
indicated]
of the
h
inverse
hyperbolic
functions
are
multiple-valued
as inverse
in the
other
inverse
hyperbolic
functions.
The
inverse
hyperbolic
functions
are
multiple-valued
and.
as
in
the and.
other inverse hyperbolic functions.
Line
ANALYTIC GEOMETRY
line
y
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0,
𝑦 = 𝑚𝑥 + 𝑏 (m-slope, b- y intercept)
𝑥 𝑦
+ =1
𝑎 𝑏
Planeb
x
a
Plane
z
𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷 = 0
Circle
c
𝑥 𝑦 𝑧
+ + =1
𝑎 𝑏 𝑐
b
a
y
x
Circle
y
(𝑥 − 𝑥8 )? + (𝑦 − 𝑦8 )? = 𝑅?
R
(𝑥𝑜 , 𝑦𝑜)
Ellipse
x
Ellipse
(𝑥 − 𝑥8 )?
𝑦 − 𝑦8
+
?
𝑎
𝑏?
y
?
=1
2a
("# , %# )
2b
x
Parabola
y
𝑦 − 𝑦8
?
− 4𝑎 𝑥 − 𝑥8 = 0
a
Focus
(𝑥𝑜 , 𝑦𝑜 )
x
Hyperbola
(𝑥 − 𝑥8 )?
𝑦 − 𝑦8
−
?
𝑎
𝑏?
?
=1
(𝑥8 , 𝑦8 ) – center
2a, 2b –major and minor axes
4
DERIVATIVES
𝑦=𝑓 𝑥
𝑑𝑦
𝑓 𝑥 + ∆𝑥 − 𝑓 𝑥
= lim
𝑑𝑥 ∆m→s
∆𝑥
; 𝑦 = 𝑓 𝑥, 𝑦
𝜕𝑓
𝑓 𝑥 + ∆𝑥, 𝑦 − 𝑓(𝑥, 𝑦)
= lim
𝜕𝑥 ∆m→s
∆𝑥
𝑑𝑢
𝑑𝑣
𝑣
− 𝑢( )
𝑑 𝑢
𝑑𝑥
𝑑𝑥
=
𝑑𝑥 𝑣
𝑣?
𝑑
𝑑𝑣
𝑑𝑢
𝑢𝑣 = 𝑢
+𝑣
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑑𝑓(𝑢) 𝑑𝑓 𝑑𝑢
=
(𝑐ℎ𝑎𝑖𝑛 𝑟𝑢𝑙𝑒)
𝑑𝑥
𝑑𝑢 𝑑𝑥
𝑑
sin 𝑥 = cos 𝑥
𝑑𝑥
𝑑
cos 𝑥 = − sin 𝑥
𝑑𝑥
𝑑
1
tan 𝑥 =
𝑑𝑥
cos ? 𝑥
𝑑
1
cot 𝑥 = − ?
𝑑𝑥
sin 𝑥
𝑑
sinh 𝑥 = cosh 𝑥
𝑑𝑥
𝑑
cosh 𝑥 = sinh 𝑥
𝑑𝑥
𝑑
1
tanh 𝑥 =
𝑑𝑥
cosh? 𝑥
𝑑
1
coth 𝑥 = −
𝑑𝑥
sinh? 𝑥
Taylor Series
𝑓 𝑥 = 𝑓 𝑎 + 𝑓… 𝑎 𝑥 − 𝑎 +
𝑑
𝑐𝑥 r = 𝑛𝑐𝑥 rpE
𝑑𝑥
𝑑
1 𝑑𝑢
ln 𝑢 =
𝑑𝑥
𝑢 𝑑𝑥
𝑑 „
𝑑𝑢
𝑒 = 𝑒„
𝑑𝑥
𝑑𝑥
𝑑 „
𝑑𝑢
„
𝑎 = 𝑎 ln 𝑎
𝑑𝑥
𝑑𝑥
𝑓′′(𝑎)(𝑥 − 𝑎)? 𝑓′′′(𝑎)(𝑥 − 𝑎)O
+
+⋯
2!
3!
INTEGRALS
𝑢 𝑥 𝑣′(𝑥) 𝑑𝑥 = 𝑢𝑣 − 𝑣 𝑥 𝑢… 𝑥 𝑑𝑥
Integration by parts:
𝑥 rlE
𝑛+1
sin 𝑥 = − cos 𝑥
sinh 𝑥 = cosh 𝑥
𝑥 r 𝑑𝑥 =
cos 𝑥 = sin 𝑥
cosh 𝑥 = sinh 𝑥
𝑑𝑥
= ln 𝑥
𝑥
tan 𝑥 = −ln cos 𝑢
tanh 𝑥 = ln cosh 𝑥
𝑒 m 𝑑𝑥 = 𝑒 m
cot 𝑥 = ln sin 𝑥
coth 𝑥 = ln sinh 𝑥
ln 𝑥 𝑑𝑥 = 𝑥 ln 𝑥 − 𝑥
Mean Value Theorem
k
𝑓 𝑥 𝑑𝑥 = 𝑏 − 𝑎 𝑓 𝑐 𝑓𝑜𝑟 𝑎 < 𝑐 < 𝑏
j
Leibnitz Rule
𝑑
𝑑𝑥
k(m)
j(m)
𝑑𝑏
𝑑𝑎
𝑓( 𝑥, 𝑡)𝑑𝑡 = 𝑓 𝑥, 𝑏
− 𝑓 𝑥, 𝑎
+
𝑑𝑥
𝑑𝑥
k(m)
j(m)
𝜕𝑓(𝑥, 𝑡)
𝑑𝑡
𝜕𝑥
5
ORDINARY DIFFERENTIAL EQUATIONS
Linear 1st Order
𝑑𝑦
+𝑃 𝑥 𝑦 =𝑅 𝑥
𝑑𝑥
𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑛𝑔 𝐹𝑎𝑐𝑡𝑜𝑟 = 𝑒
𝑑
𝑦(𝑥)𝑒 •Œm = 𝑅 𝑥 𝑒
𝑑𝑥
𝑦 𝑥 = 𝑒p
•Œm
𝑅𝑒
•Œm
𝑑𝑥 + 𝑐
‹Œm
‹Œm
Bernoulli’s Equation
𝑑𝑦
+ 𝑃 𝑥 𝑦 = 𝑅(𝑥)𝑦 r
𝑑𝑥
𝑦(𝑥)Epr = (1 − 𝑛)𝑒 p(Epr)
For n=1, ln 𝑦 =
Linear, Homogeneous,
2nd Order
𝑑? 𝑦
𝑑𝑦
+𝑎
+ 𝑏𝑦 = 0
?
𝑑𝑥
𝑑𝑥
a, b – real constants
•Œm
𝑅 𝑒 (Epr)
•Œm
𝑑𝑥 + 𝑐
𝑅 − 𝑃 𝑑𝑥 + 𝑐
𝑚E 𝑎𝑛𝑑 𝑚? 𝑎𝑟𝑒 𝑟𝑜𝑜𝑡𝑠 𝑜𝑓 𝑚? + 𝑎𝑚 + 𝑏 = 0
when 𝑚E 𝑎𝑛𝑑 𝑚? 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙
𝑦 𝑥 = 𝐶E 𝑒 gŽ m + 𝐶? 𝑒 g• m
when 𝑚E = 𝑚? = 𝑚 𝑎𝑛𝑑 𝑖𝑠 𝑟𝑒𝑎𝑙
𝑦 𝑥 = 𝐶E 𝑒 gm + 𝐶? 𝑥𝑒 gm
when 𝑚E = 𝑝 + 𝑖𝑞, 𝑚? = 𝑝 − 𝑖𝑞
𝑦 𝑥 = 𝑒 ‹m 𝐶E sin 𝑞𝑥 + 𝐶? cos 𝑞𝑥
Linear, nonHomogeneous, 2nd Order
𝑑? 𝑦
𝑑𝑦
+
𝑎
+ 𝑏𝑦 = 𝑅(𝑥)
𝑑𝑥 ?
𝑑𝑥
a,b – real constants
Same cases for 𝑚E and 𝑚? as above
𝑒 gŽ m
𝑦 𝑥 = 𝐶E 𝑒 gŽ m + 𝐶? 𝑒 g• m +
𝑚E − 𝑚?
𝑦 𝑥 = 𝐶E 𝑒 gm + 𝐶? 𝑥𝑒 gm + 𝑥𝑒 gm
𝑒 pgŽ m 𝑅𝑑𝑥 +
𝑒 pgm 𝑅𝑑𝑥 − 𝑒 gm
𝑒 g• m
𝑚? − 𝑚E
𝑥𝑒 pgm 𝑅𝑑𝑥
𝑦 𝑥 = 𝑒 ‹m 𝐶E sin 𝑞𝑥 + 𝐶? cos 𝑞𝑥
𝑒 ‹m sin 𝑞𝑥
𝑒 ‹m cos 𝑞𝑥
+
𝑒 p‹m 𝑅 cos 𝑞𝑥 𝑑𝑥 −
𝑞
𝑞
Euler Equation
𝑑? 𝑦
𝑑𝑦
𝑥 ? ? + 𝑎𝑥
+ 𝑏𝑦 = 𝑅(𝑥)
𝑑𝑥
𝑑𝑥
𝑒 pg• m 𝑅𝑑𝑥
𝑒 p‹m 𝑅 sin 𝑞𝑥 𝑑𝑥
Substitute 𝑥 = 𝑒 ’ then the equation becomes
𝑑? 𝑦
𝑑𝑦
+ (𝑎 − 1)
+ 𝑏𝑦 = 𝑅(𝑒 ’ )
?
𝑑𝑡
𝑑𝑡
See solutions above for 2nd order equations.
If 𝑅 𝑥 = 0 use 𝑦 𝑥 = 𝑥 g and follow Homogeneous 2nd Order solution
6
VECTOR
ANALYSIS
VECTOR
ANALYSIS
VECTOR ANALYSIS
DotProduct
Product
Dot
Dot
Product
|&||'|cos
cos𝜃 + 0 0≤≤𝜃 +≤≤𝜋 /
𝐀!∙ 𝐁∙ #==|𝐴||𝐵|
! ∙ # = |&||'|
cos
+ 10 ≤ + ≤ /
? 1++
?1
?
𝐴&== 𝐴m&
0 𝐴&
— 1++𝐴&
˜3
1
1
& = &0 + &1 + &13
@
7
@
7
CrossProduct
Product
Cross
Cross
Product
!×#==𝐴&|𝐵|
|'|sin
sin𝜃 +𝐧7 0 0≤≤𝜃 +≤≤𝜋 /
𝐀×𝐁
!×# = & |'| sin + 7 0 ≤ + ≤ /
𝐢
𝐣
𝐤
8
8⃗
!
=
&
;
+
&
?
+
𝐀×𝐁 = 𝐴0m 𝐴—1 𝐴˜&3 =
8⃗ = &0 ; + &1 ? + &3 =
!
𝐵 𝐵 𝐵
m
—
+
+
z
z !
8⃗ = &: ; + &> ? + &< =
88⃗
!
&< =
8!⃗
&< =
k
j
k
y
j & ?
>
i
&: ; y
&> ?
i
&: ;
88⃗
#
8⃗
#
88⃗
!
8!⃗
x x
𝐀×𝐁 = −𝐁×𝐀
𝐀 ∙ 𝐁 = 𝐴m 𝐵m + 𝐴— 𝐵— + 𝐴˜ 𝐵˜
˜
Derivative
𝑑𝐹—
𝑑𝐅 𝑑𝐹m
𝑑𝐹˜
=
𝐢+
𝐣+
𝐤
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝑑𝑥
Del Operator
𝜕
𝜕
𝜕
∇= 𝐢
+𝐣
+𝐤
𝜕𝑥
𝜕𝑦
𝜕𝑧
Gradient of a scalar
𝜕𝜙
𝜕𝜙
𝜕𝜙
∇𝜙 =
𝐢+
𝐣+
𝐤
𝜕𝑥
𝜕𝑦
𝜕𝑧
Divergent of a Vector
𝜕
𝜕
𝜕
𝜕𝐹m 𝜕𝐹— 𝜕𝐹˜
∇∙𝐅= 𝐢
+𝐣
+𝐤
∙ 𝐹m 𝐢 + 𝐹— 𝐣 + 𝐹˜ 𝐤 =
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
Curl of a Vector
𝐢
𝜕
𝜕
𝜕
𝜕
∇×𝐅 = 𝐢
+𝐣
+𝐤
× 𝐹m 𝐢 + 𝐹— 𝐣 + 𝐹˜ 𝐤 =
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝐹m
Laplacian
∇ ∙ ∇𝜙 = ∇? 𝜙 =
𝜕?𝜙 𝜕?𝜙 𝜕?𝜙
+
+
𝜕𝑥 ? 𝜕𝑦 ? 𝜕𝑧 ?
∇× ∇𝜙 = 0
𝐣
𝜕
𝜕𝑦
𝐹—
𝜕?𝐅 𝜕?𝐅 𝜕?𝐅
∇ 𝐅= ?+ ?+ ?
𝜕𝑥
𝜕𝑦
𝜕𝑧
?
∇ ∙ ∇×𝐅 = 0
̂
𝒏
Divergence Theorem
∇ ∙ 𝐅 𝑑𝑉 =
V
𝐤
𝜕
𝜕𝑧
𝐹˜
𝐅 ∙ 𝑑𝐒
S
C
S
V
S
(𝛁×𝐅) ∙ 𝑑𝐒
S
S
dS
V
Stokes Theorem
𝐅 ∙ 𝑑𝒓 =
̂
𝒏
dS
S
C
dS
̂
𝒏
dS
̂
𝒏
C
7
COORDINATE SYSTEMS
Cartesian
z
%
$
𝜕
𝜕
𝜕
∇= 𝐢
+𝐣
+𝐤
𝜕𝑥
𝜕𝑦
𝜕𝑧
P(x,y,z)
#̂
!̂
∇? Φ =
𝜕?Φ 𝜕?Φ 𝜕?Φ
+
+
𝜕𝑥 ? 𝜕𝑦 ? 𝜕𝑧 ?
z
y
Po;ar
x
y
x
Cylindrical
̂𝒛
𝒆
z
𝑥 = 𝑟 cos 𝜃
𝑦 = 𝑟 sin 𝜃
𝑧=𝑧
̂𝜽
𝒆
𝑃(𝑟, 𝜃, 𝑧)
̂𝒓
𝒆
Spherical
∇= 𝒆𝒓
∇? Φ =
𝜕
1 𝜕
𝜕
+ 𝒆𝜽
+ 𝒆𝒛
𝜕𝑟
𝑟 𝜕𝜃
𝜕𝑧
z
𝜕 ? Φ 1 𝜕Φ 1 𝜕 ? Φ 𝜕 ? Φ
+
+
+
𝜕𝑟 ? 𝑟 𝜕𝑟 𝑟 ? 𝜕𝜃 ? 𝜕𝑧 ?
x
𝜽
y
r
z
!#"
Spherical
𝑥 = 𝑟 sin 𝜃 cos 𝜙
𝑦 = 𝑟 sin 𝜃 sin 𝜙
𝑧 = 𝑟 cos 𝜃
!#$
'
&
∇= 𝒆𝒓
∇? Φ =
𝜕
1 𝜕
1
𝜕
+ 𝒆𝜽
+ 𝒆𝝓
𝜕𝑟
𝑟 𝜕𝜃
𝑟 sin 𝜃 𝜕𝜙
!#%
((*, ', &)
r
y
x
1 𝜕 ? 𝜕Φ
1
𝜕
𝜕Φ
1
𝜕?Φ
𝑟
+
sin
𝜃
+
𝑟 ? 𝜕𝑟
𝜕𝑟
𝑟 ? sin 𝜃 𝜕𝜃
𝜕𝜃
𝑟 ? 𝑠𝑖𝑛? 𝜃 𝜕𝜙 ?
8