曲线坐标系,delta函数

A vector is expressed in terms of its three components and the unit vectors of x,y,z axes.
ˆ ˆ Spherical: A Ar r A A ? ˆ
The axes in spherical coordinate system are not defined yet, let alone the unit vectors of them. But we do need the unit vectors of the coordinate axes in order to express a vector. YUNNAN NORMAL UNIVERSITY
Infinitesimal volume element in spherical coordinate system
Cartesian
x dx, y dy, z dz
Spherical
r dr, d , d
ˆ z
dsz
ˆ r
ˆ y
dsy
dsr
ds
ds
ˆ ˆ zdS zdxdy ˆ ˆ ydS ydxdz ˆ ˆ xdS xdydz
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dsx
ds
ds
ຫໍສະໝຸດ Baidu
ˆ
ˆ
ˆ ˆ rdS rds ds ˆr 2 sin dd r
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Spherical coordinate system
Construct spherical coordinate system with the help of Cartesian coordinate system In Cartesian coordinate system
Parameters x,y,z are used to specify a point.
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How to represent a vector in spherical coordinate system?
ˆ ˆ Cartesian: A Ax x Ay y Az z ˆ
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Coordinate System
Preoccupational Pitfalls
1. A coordinate system should consist of one origin and three coordinate axes. 2. The directions of the coordinate axes remain constant throughout the space.
Every simplification has a cost. The formulism under curvilinear system is more complicated.
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Spherical coordinate system
Define a system that can uniquely describe a point in the space:
x r sin cos y r sin sin z r cos
ˆ ˆ zr ˆ ˆ ˆ x sin y cos sin
ˆ ˆ ˆ r ˆ ˆ ˆ x cos cos y cos sin z sin
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The definition of coordinate axes in spherical coordinate system
Infinitesimal area element in spherical coordinate system
Cartesian
x dx, y dy, z dz
Spherical
ˆ z
dsz
r dr, d , d
ˆ y
dsy
ˆ r
ˆ x
dsr
x, y, z
Curvilinear Coordinate System & Delta Function
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1
Coordinate System
What’s the use of coordinate systems?
Find out the relationship between (x,y,z) and r , ,
Spherical coordinate system
Parameters r , , are used to specify a point. YUNNAN NORMAL UNIVERSITY
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Infinitesimal displacement in spherical coordinate system
The geometric way: Let dsr , ds , ds be the distance changes along three axes
The derivatives of the unit vectors
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The derivatives of the unit vectors
ˆ ˆ ˆ x y z 0, k, h, s are arbitrary variables Cartesian system: k h s
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Spherical coordinate system
Are the coordinate axes fixed all the time?
The directions of coordinate axes depend on the location of the point. Whereas in Cartesian coordinate system the directions of axes are always fixed.
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Infinitesimal displacement in spherical coordinate system
ˆ ˆ ˆ Cartesian system: ds xdx ydy zdz
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Why Curvilinear Coordinate System?
Why do we have different shapes of screw driver heads? Different problems need different tools. Cartesian system is simple to understand, but is not always easy to apply. Recall some problems in classical electromagnetism where curvilinear systems can best exploit the symmetry.
Examples of Coordinate System
Cartesian coordinate system: the simplest
ˆ z
ˆ y
ˆ x
Cartesian coordinate system
Spherical coordinate system
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Uniquely determine the position of a point by using one or more numbers (coordinates)
The basic elements of a coordinate system:
The origin The “system” which tells you how to uniquely specify the position of a point with a group of numbers in reference to the origin
ˆ ˆ ˆ ds rdsr ds ds
dsr dr
ds rd
ds r sin d
ˆ ˆ ˆ ds rdr rd r sin d
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ˆ
x, y, z
dV dsx dsy dsz dxdydz
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dsx
ˆ x
ˆ
dV dsr ds ds r 2 sin drdd
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What are the coordinate axes?
Geometric definition:
ˆ ˆ ˆ r xx yy zz ˆ r r r ˆ ˆ ˆ x sin cos y sin sin z cos
ˆ r
ˆ
ˆ
Abandon the prejudice that coordinate axes’ directions are fixed!
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The directions of axes are not always fixed!
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Coordinate axes are not necessary when determining a point in the space. And their directions are not always fixed.
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What about spherical system? There are two ways of obtaining infinitesimal displacement: The algebraic way:
ˆ ˆ ˆ ds d (rr ) rdr rdr ˆ ˆ ˆ r r r ˆ rdr r ( dr d d ) r ˆ ˆ ˆ rdr rd r sin d