Lecture-Differential-Length-Area-Volume.pdf - EMPossible

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Electromagnetics:

Electromagnetic Field Theory

Differential Length, Area & Volume

Outline

•Cartesian Coordinates•Cylindrical Coordinates• Spherical Coordinates

Slide 2

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Slide 3

Cartesian Coordinates

Cylindrical Coordinates

Spherical Coordinates

Cartesian Coordinates

Slide 4

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Cartesian: Differential Lengths

Slide 5

ˆxd dxa


Slide 6

ˆ ˆx yd dxa dya

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Slide 7

ˆ ˆ ˆx y zd dxa dya dza

Cartesian: Differential Areas

Slide 8

ˆxds dydza

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Slide 9

ˆ ˆx yds dydza dxdza


Slide 10

ˆ ˆ ˆx y zds dydza dxdza dxdya

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Cartesian: Differential Volume

Slide 11

dv dxdydz

Cylindrical Coordinates

Slide 12

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Cylindrical: Differential Lengths

Slide 13

ˆd d a

d


Slide 14

ˆ ˆd d a d a

d

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Slide 15

ˆ ˆ ˆzd d a d a dza

d

Cylindrical: Differential Areas

Slide 16

ˆds d dza

d

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Slide 17

ˆ ˆds d dza d dza

d


Slide 18

ˆ ˆ ˆzds d dza d dza d d a

d

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Cylindrical: Differential Volume

Slide 19

dv d d dz

d

Spherical Coordinates

Slide 20

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ˆrd dra

Spherical: Differential Lengths

Slide 21

ˆrdrard

dr

rd

sinr d


Slide 22

ˆrdraˆsinr d a

ˆ

ˆsinrd dra

r d a

rd

dr

rd

sinr d

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Slide 23

rd

dr

rd

ˆ

ˆ

ˆsin

rd dra

rd a

r d a

ˆrdra

ˆrd a

ˆsinr d a

sinr d

Spherical: Differential Areas

Slide 24

2 ˆsin rds r d d a

2 ˆsin rr d d a

dr

dr

rd

sinr d

rd

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Slide 25

2 ˆsin

ˆsinrds r d d a

r drd a

2 ˆsin rr d d a

ˆsinr drd a

dr

dr

rd

sinr d

rd


Slide 26

2 ˆsin

ˆsin

ˆ

rds r d d a

r drd a

rdrd a

2 ˆsin rr d d a

ˆsinr drd a

ˆrdrd a

dr

dr

rd

sinr d

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Spherical: Differential Volume

Slide 27

2 sindv r drd d

dr

rd

dr

rd

sinr d

Lecture-Differential-Length-Area-Volume.pdf - EMPossible

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