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An overview of the concepts of derivatives and integrals in the context of multivariable functions. It covers topics such as partial derivatives, indefinite and definite integrals, and the application of these concepts in various coordinate systems like cartesian, spherical, and cylindrical. The document also discusses special derivatives like divergence and rotation, as well as important theorems like stokes' theorem and green's theorem. Additionally, it explores charge distributions, potential, and energy in electromagnetic fields. This comprehensive coverage of derivatives and integrals in multivariable functions can be valuable for students studying advanced mathematics, physics, or engineering disciplines.
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th
ed.), William H. Hayt,
John A. Buck.
Subject contents
1.3. Indefinite integral, definite integral
F x( ) f x dx F ( )x f x
F x y
F x y f x y dx f x y
x
2
ax b dx ax bx C
2
1
2
2
x y dx x yx C
x y dy xy y C
1.3. Indefinite integral, definite integral
b
b
a
a
f x dx F x F b F a
F x f x
with
Example:
2 2
2
1 1
ax b dx ax bx C a b C a b C a b
1.3. Indefinite integral, definite integral
b
b
a
a
f x y dx F x y F b y F a y
F x y
f x y
x
with
Example:
2
2
2
1 1 1
1
1
2 2
2
2 2 2
1 1
x y dx x yx C y C y C y
x y dy xy y C x C x C x
1.4. Multi-level integrals
Double integral:
2
S OAB
2 3
1
1 2
x y
I x y dx dy
1.4. Multi-level integrals
2
V OABC
I x y dx dy dz A B C
with
2 3 1
1
1 2 1
x y z
I x yz dx dy dz
1.5. Vector, vector function
1.5. Vector, vector function
AB
AB
1.5. Vector, vector function
AB AC AC CD AD
AB AC CB since AC CB AB
1.5. Vector, vector function
2
, , sin
x y z
example :E x y z x a xy a z a
1.6. Basic coordinate systems
1.6.1. Cartesian coordinate system
1.6.1. Cartesian coordinate system
x
y
z
x
y
z
x
y
z
1.6.1. Cartesian coordinate system
x y z x y z
y
y
y y
A A A A A A A dy A
Oy Oy
AA dy
dl dy
d dy
AA a
l a
1.6.1. Cartesian coordinate system
x y z x y z
z
z
z z
A A A A A A A A dz
Oz Oz
AA dz
dl dz
d dz
AA a
l a
1.6.1. Cartesian coordinate system
x y y z z x
x x
x y
x y z
a a a a a a
a a
a a
a a a
1.6.1. Cartesian coordinate system
x x y y z z
d l dl a dl a dl a
1.6.1. Cartesian coordinate system
z x y
y
d dl dl
dz dx
s a
a
1.6.1. Cartesian coordinate system
x y z
z
d dl dl
dx dy
s a
a
1.6.1. Cartesian coordinate system
x y z
dv dl dl dl
dx dy dz
1.6.2. Spherical coordinate system
r 0,
1.6.2. Spherical coordinate system
1.6.2. Spherical coordinate system
2
sin
sin
r
r
r
d dl dl
r d r d
r d d
s a
a
a
1.6.2. Spherical coordinate system
sin
sin
r
d dl dl
dr r d
r dr d
s a
a
a
1.6.2. Spherical coordinate system
r
d dl dl
dr r d
r dr d
s a
a
a
