16.360 Lecture 13

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16.360 Lecture 13
Basic Laws of Vector Algebra
Scalars
e.g. 2 gallons, 1,000, 35ºC
Vectors
e.g. velocity 35mph heading south 3N
force toward center
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16.360 Lecture 13

• Cartesian coordinate system

z
A
?
y
?
x
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16.360 Lecture 13

• Vector addition and subtraction

C BA A B,
A
C
C
parallelogram rule
A
head-to-tail rule
C BA A B,
B
B
D A - B -(B A),
A
A
D
D (B-A)
B
B
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• position and distance

z
A
B
y
D A - B -(B A),
x
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• Vector multiplication

1. simple product
2. scalar product (dot product)
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Properties of scalar product (dot product)
a) commutative property
b) Distributitve property
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3. vector product (cross product)
a) anticommutative property
b) Distributitve property
c)
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3. vector product (cross product)
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Example vectors and angles

• In Cartesian coordinate, vector A is directed
  from origin to point P1(2,3,3), and vector B is
  directed from P1 to pint P2(1,-2,2). Find
• (a) Vector A, its magnitude A, and unit vector
  a
• (b) the angle that A makes with the y-axis
• (c) Vector B
• (d) the angle between A and B
• (e) perpendicular distance from origin to vector
  B

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4. Scalar and vector triple product
a) scalar triple product
b) vector triple product
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Example vector triple product
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• Cartesian coordinate system

z
dl
?
y
?
x
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• Cartesian coordinate system

z
directions of area
y
x
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• Cylindrical coordinate system

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• the differential areas and volume

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Example cylindrical area
z
5
3
y
60
30
x
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• Spherical coordinate system

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• differential volume in Spherical coordinate
  system

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• Examples

(1) Find the area of the strip
(2) A sphere of radius 2cm contains a volume
charge density
Find the total charge contained in the sphere
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• Cartesian to cylindrical transformation

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• Cartesian to cylindrical transformation

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• Cartesian to cylindrical transformation

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• Cartesian to Spherical transformation

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• Cartesian to Spherical transformation

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• Cartesian to Spherical transformation

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• Distance between two points

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16.360 Lecture 16
Gradient in Cartesian Coordinates
Gradient differential change of a scalar

The direction of
is along the maximum increase of T.
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Example of Gradient in Cartesian Coordinates
Find the directional derivative of
along the direction
and evaluate it at (1, -1,2).
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Gradient operator in cylindrical Coordinates
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Gradient operator in cylindrical Coordinates
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Gradient operator in Spherical Coordinates
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Properties of the Gradient operator
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Flux in Cartesian Coordinates
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Flux in Cartesian Coordinates
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Definition of divergence in Cartesian Coordinates
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Properties of divergence
If
No net flux on any closed surface.
Divergence theorem
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Divergence in Cylindrical Coordinates
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Divergence in Cylindrical Coordinates
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Divergence in Spherical Coordinates
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Divergence in Spherical Coordinates
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Divergence in Spherical Coordinates
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16.360 Lecture 18
Circulation of a Vector
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Circulation of a Vector
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Curl in Cartesian Coordinates
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Vector identities involving the curl
Stokess theorem
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Curls in Rectangular, Cylindrical and Spherical
Coordinates
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Laplacian Operator of a scalar
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Laplacian Operator of a vector