Vector Analysis - LectureNotes2014.05 - Differential Length, Surface and Volume (1)

Vector Analysis 8/19/2014

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Fernando Victor V. de Vera, ECE, M.Tech

[email protected]

[email protected]

University of the East - Manila Campus

College of Engineering

Electronics Engineering Department

DIFFERENTIAL DISPLACEMENT, AREA AND VOLUME


Differential Length

�CARTESIAN

�� = �� + �� + ��

NOTE: Differential Length

element is a vector quantity

z

y

x

dz

dy

dx

az

ax ay


Differential Length

�CYLINDRICAL CORDINATES

�� = �ρ� + ρ�φ�� + ��

z

y

x

φ

ρ

dρ

ρ dφ

dz


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Differential Length

�SPHERICAL

�� = �� + ��θ�� + ��θ��

z

y

x

φ

r θ

r dθ

dr

r sinθ dφ


Example No. �Based on the figure,

let point Q be at the

origin, B(5, 80o, 2),

C(5, 15o, 2), find the

distance from:

a. C to B

b. A to R

c. B to S

A

B

C

Q

R

S


Example No. � Consider a portion of a

spherical volume with the following points as follows: A(3, 10o, 20o), C(7, 10o, 75o) and G(7, 100o, 75o). Find the straight line distance from: a) B to D

b) C to E

� Find the distance following the curvature of the surface from:

c) B to E

d) C to H


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Practice Exercises No. �Find the total edge length of each of the

regions described as follows:

a) 0 < x,y,z < 7

b) 2 < ρ < 5; 1.1π < φ < 1.8 π; -3 < z < 3

c) 0 < r < 9; 0o < θ <160o; 120o < φ < 200o


Differential Surface

CARTESIAN

�� = ��

�� = ��

�� = ��

y

x

dz

dy

dx

az

ax ay



CYLINDRICAL

�� = ��

�� = ��

�� = ��

z

y

x

φ

ρ

dρ

ρ dφ

dz


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SPHERICAL

�� = � �� θ ��

�� = � ��θ �� φ ��

�� = �� θ �θ �φ ��

z

y

x

φ

r θ

r dθ

dr

r sinθ dφ


Example No. �Based on the figure,

let point Q be at the

origin, B(5, 80o, 2),

C(5, 15o, 2), find the

area of the following:

a. ABC

b. BCRS

A

B

C

Q

R

S


Example No. �Consider a portion of a

spherical volume with the following points as follows: A(3, 10o, 20o), C(7, 10o, 75o) and G(7, 100o, 75o). Find the surface area of the following:

a. ABCD

b. CDGH


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Practice Exercises No. �Find the total surface area of each of the

regions described as follows:

a) 0 < x,y,z < 7

b) 2 < ρ < 5; 1.1π < φ < 1.8 π; -3 < z < 3

c) 0 < r < 9; 0o < θ <160o; 120o < φ < 200o


Differential Volume

�Differential volume is a scalar quantity.

CARTESIAN

�� = ��

CYLINDRICAL

�� = ρ�ρ�φ�

SPHERICAL

�� = ��θ��θ�φ


Example No. �Consider a portion of

a spherical volume

with the following

points as follows: A(3,

10o, 20o), C(7, 10o,

75o) and G(7, 100o,

75o). Find the volume.


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Practice Exercises No. �Find the volume of the region described as

follows:

a) 0 < x,y,z < 7

b) 2 < ρ < 5; 1.1π < φ < 1.8 π; -3 < z < 3

c) 0 < r < 9; 0o < θ <160o; 120o < φ < 200o


Practice Exercises No. �A portion of a hollow cylindrical tube has a

length of 5units. The inner and outer radius is

3 and 6 units respectively. If 20o<φ<330o:

a) Sketch the region

b) Find the points of all corners

c) Find the total surface area

d) Find the total volume


Practice Exercises No. �For the spherical region defined as 4<r<5;

150o<θ<170o; 0<φ<2π:

a) Sketch the region

b) Compute for the total edge length

c) Compute for the total surface area

d) Compute for the volume

Vector Analysis - LectureNotes2014.05 - Differential Length, Surface and Volume (1)

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