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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 3-dimensional 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

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.

Unit vectors: ix, iy, iz (also called ax, ay, az)

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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, volumesUnit 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 hand if thumb along z axis.), θ = angle made with line drawn out to the point and the positive z axis

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Differential lengths, areas, volumes

Unit vectors: ir, iФ, iθ (mutually orthogonal)Position vector: rir

Differential lengths: dr, rsinθdФ, rdθDifferential areas: rsinθdrdФ, rdrdθ, r2sinθdθdФDifferential volume: r2sinθdФdθdrCoordinate 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.