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An in-depth exploration of vectors and scalars, including their definitions, unit vectors, magnitudes, and vector algebra. It covers the concepts of dot product and cross product, as well as vector transformations in rectangular, cylindrical, and spherical coordinates. The document also includes formulas and explanations for coordinate systems, differential surfaces, and volumes.
Typology: Study notes
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A vector is a quantity having both magnitude and direction. A scalar is a quantity only having magnitude. A unit vector has magnitude one and is used to denote direction.
To find the unit vector of vector A:
2 2 2 x y z
A
x x y y z z
Projection
To find out how much of vector B is in the direction of a :
A B B A
a a a A B
A B A B a
× =− ×
× =
× =
x y z
x y z
x y z
AB n
B B B
A A A
sin θ
Expand this determinant
dydx a z
dx dz a y
dy dz a x
Volume = dx dy dz
Surfaces in Rectangular Coordinates
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
Surfaces in Cylindrical Coordinates
dr
r sinθ dφ r dθ
r dθ dr aφφ
r sinθ dφ dr a θ
r sinθ dφ r dθ a r
dv= r sinθ dφ r dθ dr
Surfaces in Spherical Coordinates
Given:
A =Ax ax + Ay ay + Az az (rectangular)
Find:
A =Aρ aρρ + Aφ aφφ + Az az (cylindrical)
We must use the projection property
Aρ = A •aρρ
Aφ = A •aφφ
Az = A •az
Vector Transformation: Cylindrical
What is φ? use equations on page 17 #
Cylindrical Point transformations:
x= r cosφ
y= r sinφ
z= z
r = sqrt(x^2 +y^2 )
φ= tan-1^ y / x z= z
Note: A vector transformation is always done at a point
A = Ax ax + Ay ay + Az az
A = Ar ar + Aθ aθθ + Aφ aφφ
Ar = A• ar
Aθ = ΑΑ • aθθ
Aφ = A • aφφ
Then use table 1.2 for unit vector products.
!!For θ, φ use point transformations at some point (r, θ, φ)
