Lecture 4
Coordinate Systems:
Rectangular, Cylindrical,
Spherical
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We will use all three of these
coordinate systems to represent
the position of a point P in 3dimensional space.
In all three cases, position is
defined in terms of the
intersection of 3 surfaces.
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In each coordinate system, we shall
define three coordinate variables
and three corresponding “unit
vectors”. Each unit vector has unit
length, and points in the direction of
increasing value of the coordinate
variable to which it corresponds.
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All three coordinate systems are
“mutually orthogonal”. This
means that their three unit vectors
are mutually perpendicular. This
makes it easy to calculate dot and
cross vector products. (Eg: All dot
products between pairs of unit
vectors are zero!)
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Rectangular (Cartesian) Coordinates
Point P(x,y,z) is defined as intersection of 3 planes: plane of constant x, plane of constant y, and plane of constant z
x = distance from y-z plane, y = distance from x-z plane, z = distance from x-y plane
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Differential lengths, areas, volumes
in Rectangular coordinate system
Unit vectors: ix, iy, iz (also called ax, ay, az)
Position vector drawn from origin to point (x,y,z)
x ix + y iy + z iz
Differential lengths
dx, dy, dz
Differential areas
dxdy, dxdz, dydz
Differential volume
dxdydz
Variable range
x,y,z take on all values
Note that the unit vectors are mutually orthogonal
Unit vector direction remain constant at all
positions in space.
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Cylindrical Coordinate System
Point P(r,Ф,z) defined as intersection of two planes and a cylinder: plane of constant z,
plane of constant Ф, and cylinder of constant r (or ρ)
r (or ρ) = distance from z axis, Ф = angle from positive x axis (positive wrt to
fingers of right hand if thumb along z axis.) z = distance from x-y plane
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Differential lengths, areas, volumes
Unit vectors: ir, iФ, iz (mutually orthogonal)
Position vector: r ir + z iz
Differential lengths: dr, rdФ, dz
Differential areas: rdФdr, drdz, rdФdz
Differential volume: rdФdrdz
Coordinate range: r = (0 to infinity),
Ф = (0 to 2π)
z = (-infinity to +infinity)
Useful for systems with cylindrical symmetry, but the iФ
and ir unit vectors do not point in constant directions,
but rather, their direction is a function of position.
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Spherical Coordinates
Point P(r,Ф, θ) defined as intersection of a plane, sphere, and cone… constant Ф plane,
constant r sphere, and constant θ cone.
r = distance from the origin, Ф = angle from positive x axis (positive wrt to fingers of right
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hand if thumb along z axis.), θ = angle made with line drawn out to the point and the
positive z axis
Differential lengths, areas, volumes
Unit vectors: ir, iФ, iθ (mutually orthogonal)
Position vector: rir
Differential lengths: dr, rsinθdФ, rdθ
Differential areas: rsinθdrdФ, rdrdθ, r2sinθdθdФ
Differential volume: r2sinθdФdθdr
Coordinate range: r = (0 to infinity),
Ф = (0 to 2π)
θ = (0 to π)
Useful for systems with spherical symmetry, but
none of the unit vectors have a constant direction.
The direction of each unit vector is a function of
position.
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Conversion between coordinate systems
Note that rc and rs distinguish between the r (or ρ) cylindrical
coordinate and the r spherical coordinate.
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All three coordinate systems will
be useful in this course, but we
shall choose to use rectangular
coordinates unless a compelling
cylindrical or spherical symmetry
exists in the problem.
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