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Derivatives and Integrals in Multivariable Functions, Summaries of Linear Algebra

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.

Typology: Summaries

2023/2024

Uploaded on 05/13/2024

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Fundamentals of Electromagnetics Fields
Assoc. Prof., Dr. Sc. Trần Hoài Linh
School of Electrical and Electronics Engineering
linh.tranhoai@hust.edu.vn, thlinh2000@yahoo.com
Subject contents
Slides: Materials for tests.
Text book(s): Engineering Electromagnetics (recommended 6th ed.), William H. Hayt,
John A. Buck.
Midterm test and Final test: 3 questions (9 points) + 1 point for presentation
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Download Derivatives and Integrals in Multivariable Functions and more Summaries Linear Algebra in PDF only on Docsity!

Fundamentals of Electromagnetics Fields

Assoc. Prof., Dr. Sc. Trần Hoài Linh

School of Electrical and Electronics Engineering

linh.tranhoai@hust.edu.vn, thlinh2000@yahoo.com

Subject contents

  • Slides: Materials for tests.
  • Text book(s): Engineering Electromagnetics (recommended 6

th

ed.), William H. Hayt,

John A. Buck.

  • Midterm test and Final test: 3 questions (9 points) + 1 point for presentation

Subject contents

1. Selected mathematics tools required

2. Static electrical field

3. Steady electrical field

4. Static magnetic field

5. Time-varying fields

6. Waves propagation and antennas

  1. Selected mathematics tools required
    • Single variable, multi-variable functions
    • Derivative, partial derivatives
    • Integral (indefinite, definite)
    • Multi-level integrals
    • Vectors and vector functions
    • 3 basic systems of coordinations
    • Line integral, surface integral, volume integral
    • Stokes’, Green’s theorems
    • Integral of gradient

1.3. Indefinite integral, definite integral

  • Indefinite integral: finding antiderivative function

Example:

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

  • 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

  • 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

  • Usually deal with definite integrals.
  • In “Fundamentals of Electromagnetic Fields”: usually up to 3 levels (triple

integral).

  • Example:

 Double integral:

 

 

2

S OAB

I x y dx dy

with A B



 

2 3

1

1 2

x y

I x y dx dy

 

 

1.4. Multi-level integrals

  • Triple integral:

       

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

  • Vector: defined by 4 parameters:
    • The origin (the tail): A
    • The direction: from A to B (from tail to head)
    • The magnitude: length of AB
  • Unit vector: the vector with length = 1

1.5. Vector, vector function

  • Vector: defined by 4 parameters:
    • The origin (the tail): A
    • The direction: from A to B (from tail to head)
    • The magnitude: length of AB
  • Unit vector: the vector with length = 1

AB

a

AB

AB  AB  ABa

1.5. Vector, vector function

  • Vector addition and subtraction:

AB  AC  AC  CD AD

AB  AC  CB since AC  CB AB

1.5. Vector, vector function

  • Vector function: the value of the function at a point is a vector

   

2

, , sin

x y z

example :E x y z  x  a  xy  a  z a

1.6. Basic coordinate systems

  • 3 fundamental coordinate systems:
    • Cartesian (rectangular) coordinate system
    • Spherical coordinate system
    • Cylindrical coordinate system
  • For each coordinate system:
    • 3 coordinates/components,
    • The unit vectors for each coordinate,
    • Differential displacement (line) vector,
    • Differential surface vector
    • Differential volume (scalar)

1.6.1. Cartesian coordinate system

  • 3 coordinates: x, y, z
  • 3 “axis” x-y-z form a right-hand-triple

1.6.1. Cartesian coordinate system

  • 3 coordinates: x, y, z
  • 3 “axis” x-y-z form a right-hand-triple
  • Unit vectors (for each coordinate): a

x

, a

y

, a

z

(other format (i, j, k) or (i

x

, i

y

, i

z

  • Ranges of coordiantes:

 

 

 

x

y

z

1.6.1. Cartesian coordinate system

  • Unit vectors at a given point (not only at the origin

O(0,0,0)): The unit vector for 1 coordinate has the

same direction as the differential displacement

vector at the given point for the given coordinate!

   

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

  • Unit vectors at a given point (not only at the origin

O(0,0,0)): The unit vector for 1 coordinate has the

same direction as the differential displacement

vector at the given point for the given coordinate!

   

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

  • Unit vectors at a given point (not only at the origin

O(0,0,0)): The unit vector for 1 coordinate has the

same direction as the differential displacement

vector at the given point for the given coordinate!

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

With all 3 components determined, the general

differential displacement vector is given by:

x x y y z z

d l  dl  a  dl  a  dl a

1.6.1. Cartesian coordinate system

  • For example surface with y = const

 

 

z x y

y

d dl dl

dz dx

   

   

s a

a

1.6.1. Cartesian coordinate system

  • For example surface with z = const

 

 

x y z

z

d dl dl

dx dy

   

   

s a

a

1.6.1. Cartesian coordinate system

  • Differential volume at a point (A) = product of 3 differential displacements for all

coordinates at A:

x y z

dv dl dl dl

dx dy dz

  

  

1.6.2. Spherical coordinate system

  • 3 coordinates: r, , 
  • Ranges of coordinates:
    • r – length of OA:
    •  - angle from Oz to OA:
    •  - angle from Oz to OH:

 

r  0,

 

 

1.6.2. Spherical coordinate system

  • Differential surface vector: Limit consideration for surfaces with 1 coordinate is

constant!

At those cases, the differential surface vector at a point (A) is determined as follow:

  • The direction is normal (perpendicular) to the surface at A (i.e. parallel to the unit

vector of the constant coordinate)

  • The direction is facing “outward” of the given surface
  • The magnitude equals the product of two differential displacements of the two

varying coordinates

1.6.2. Spherical coordinate system

  • For example surface with r = const

 

    

 

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

  • For example surface with  = const

 

   

 

sin

sin

r

d dl dl

dr r d

r dr d

 

 

   

     

     

s a

a

a

1.6.2. Spherical coordinate system

  • For example surface with  = const (please note

that it’s only ½ plane)

 

   

 

r

d dl dl

dr r d

r dr d

   

    

    

s a

a

a