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Unformatted text preview: ECE 107: Electromagnetism
Set 3: Review of vector analysis
Instructor: Prof. Vitaliy Lomakin
Department of Electrical and Computer Engineering
University of California, San Diego, CA 92093 1 Why do we need vector analysis?
Maxwell’s equations
Time domain Maxwell’s equations:
∂B
∂t
∂D
∇×H =
+J
∂t ∇×E = − ∇⋅D = ρ
∇⋅B = 0 Basic quantities: E(t , r ) − electric field
H(t , r ) − magnetic field Vm
Am D(t , r ) − electric flux density C m2 B(t , r ) − magnetic flux density webers m2 J − electric current density A m2 ρ − charge density Cm 3 2 Scalar fields Single value function that depends on position Vector fields A vector is given for each position in space Velocity of each atom flow of heat ˆ
ˆ
ˆ
ˆ
ˆ
A = a A = aA = x Ax + y A y + z Az ˆ
a =1 Vector fields
A vector is given for each position in space ˆ
ˆ
ˆ
ˆ
ˆ
A = a A = aA = x Ax + y A y + z Az A = A2 + A2 + A2
x
y
z ˆ
ˆ
ˆ
C = A + B = x(Ax + B x ) + y(Ay + B y ) + z(Az + B z ) C A B Vector fields
Position Vector: From origin to point P Distance Vector: Between two points Scalar and vectors products
• Scalar product • Vector product ˆ
n ⊥ A & B (righthanded rule)
7 Vector relations
cartesian cylindrical
spherical 8 Relation between the coordinates 9 Gradient
• Gradient
– Gradient applies to a scalar. Its result is a vector of
maximum rate of change of the scalar
del operator directional derivative – Del operator – Operations with the gradient 10 Divergence
• Divergence
– Divergence applies to a vector. Its result is a scalar
giving the flux per unit volume. – Gauss’s theorem:
–
11 Curl
• Curl operator
– Curl operator applies to a vector and results in a
vector. Curl is the maximal circulation of the vector
per unit area – Stockes’s theorem:
– Vector identities: 12 Laplacian
• Scalar Laplacian operator
– Applies to a scalar and produces a scalar • Vector Lapacian operator
– Applies to a vector and produces a vector
this simple only
in the Cartesian coordinates! is valid in any coordinate system
with proper expresion for ∇! 13 Gradient, divergence, and curl 14 ...
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This note was uploaded on 04/02/2012 for the course ECE 107 taught by Professor Fullterton during the Winter '07 term at UCSD.
 Winter '07
 fullterton

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