<A Student’s Guide to Vectors and Tensors> by Danil A. Fleisch
http://www.danfleisch.com/sgvt
A scalar is a single value with no directional indicator that represents a quantity that does not vary as the coordinate system is changed.
A vector is an array of three values (in 3-D space) called “vector components” that combine with directional indicators (“basis vectors”) to form a quantity that does not vary as the coordinate system is changed.
A tensor of rank n is an array of 3 n values (in 3-D space) called “tensor com-
ponents” that combine with multiple directional indicators (basis vectors) to
form a quantity that does not vary as the coordinate system is changed
向量外积：
x1, x2, x3 ---> x'1, x'2, x'3
A×(B×C)=B(A◦C)−C(A◦B).
B×C和A×(B×C)共面
▽：
▽◦：
▽×：
△φ
the Laplacian of a function represents the difference between the value of the function at that point and
the average of the values at surrounding points. (How does it do that? P55)
And how does the difference between a function’s value at a point and the average value at neighboring points relate to the divergence of the gradient of that function? P57
△：
covariant components transform in the same way as basis vectors
contravariant components transform in the opposite way to basis vectors
covariant basis vector(坐标轴) ～dx
contravariant vector component(坐标值) ～▽x
the “transformation” of basis vectors refers to :
the conversion of the basis vectors in the original (non-rotated) coordinate system to the different basis vectors which point along the coordinate axes in the new (rotated) system
the “transformation” of vector components refers to:
the change in the components of the same vector referred to two different sets of coordinate axes
v=(x, y, z), 分量x,y,z是v垂直投影到各个坐标轴上的值。所以x,y,z不能作为parallel coordinate system下v的坐标。
由此，想到的问题是：有没有其他的向量表示方法，在适用于直角坐标系的情况下，还可以推广到非直角坐标系？答案是人们发明了dual basis vectors (reciprocal basis vectors)
e$^1$: e1的dual vector
e$^2$: e2的dual vector
Dual basis vectors characteristics (2D)：
- e$^1$┴e2, e$^2$┴e1
- e$^1$◦e1=1, e$^2$◦e2=1
注意：|e$^i$| = 1/(|ei|*cos(thta)) = 1/cos(thta) != 1 (thta: e$^i$ ^ ei)
（通俗理解：A向x轴投影 ～ 向“与y轴无关”方向投影 ～ 向“与y轴垂直”的方向投影 ～ 向e$^1$投影）
A向e1上投影得A$^x$,
向e2上投影得A$^y$
A向(e1的dual vector)e$^1$上投影得Ax,
向(e2的dual vector)e$^2$上投影得Ay,
如何求dual vector：
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给定A，坐标轴e1，e2，求用e1，e2 表示A，(即，求A$^1$, A$^2$):
示意图：
如果先求出dual vector，就很简单了：
另一个方法是用matric tensor求： A $^j$ = g$^{ij}$ A$_i$ (g$_{ij}$ g$^{ij}$ = I)
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给定A，坐标轴e1，e2，求用e1,e2 的dual vector(即e$^1$, e$^2$)表示A (即求A1,A2):
示意图：
首先要求出dual vector
然后再求出A1，A2：
另一个方法是用matric tensor求： A $_i$ = g$_{ij}$ A$^j$ (g$_{ij}$ g$^{ij}$ = I)
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the ∂x‘i/ ∂xj terms are not only the elements of a transformation matrix from the unprimed to the primed coordinate system, they’re also the components of the basis vectors tangent to the orig-
inal (unprimed) coordinate axes, expressed in the new (primed) coordinate system.
they represent the components of the original (unprimed) covariant basis vectors expressed in the new (primed) coordinate system.
∂xj/ ∂x‘i are the components of the (contravariant) dual basis vectors
At this point you should be convinced that vectors are more than just lit-
tle arrows with magnitude and direction; they’re quantities that transform in
certain ways between coordinate systems.
Invariant
= vector’s contravariant components(A$^i$) + covariant basis vector(ei)
= vector’s covariant components(Ai) + contravariant basis vector(e$^i$)(1-form)
where the ∂ ∂ x x j terms represent the components in the new coordinate system
of
the basis vectors tangent to
the original axes
where the ∂ x i ∂ x terms represent the components in the new coordinate system
of the (dual) basis vectors perpendicular to the original axes.
坐标系1下的向量A变换为坐标系2下的向量A‘，A‘的各分量可以看作是A的各分量的加权之和。
同理，对于张量：
坐标系1下的张量A变换为坐标系2下的张量A‘，A‘的各分量可以看作是A的各分量的加权之和。
the elements of the (direct) transformation matrix represent the basis vectors tangent to the original coordinate axes（contravariant basis vector表示）
the elements of the transformation matrix represent the dual basis vectors perpendicular to the original coordinate axes（covariant basis vector表示）
rank-2 tensor的其他表示：
（混合表示）
tensor加法：
tensor outer-product:
the Kronecker Delta function:
That tensor, the one that “provides the metric” for a given coordinate system in the space of interest, is called the fundamental or metric tensor
tensor inner product：outer-product + contraction
在上面的outer-production结果基础上，进行contraction, 这里假设contract q和n：
上面contract q和n， 当然也可以contract其他项。
对于vector A，B，A◦B = A⊗B , 然后contract A⊗B,
》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》》
用张量的符号推导向量内积：
由于只有i和m，所以只能contract i和m
但是继续往下怎么推导?
《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《《
5.5 Metric tensor
why do you need a coordinate
system? Because coordinate systems “arithmetize” space – that is, they give
you a way of applying the rules of arithmetic to objects that exist in the space
in which you’re working.
The coordinate system you apply may have straight
axes that intersect at right angles, or the axes may be curved and intersect at
any angle of your choosing.
However you choose to arithmetize a space, there is one tensor that allows
you to define fundamental quantities such as lengths and angles in a consistent
manner at different locations. That tensor, the one that “provides the metric”
for a given coordinate system in the space of interest, is called the fundamental
or metric tensor.
Hence it must be the job of the metric tensor g ##
and its components g$^{i j}$ and g$_{i j}$ to turn the product of incremental coordinate
changes expressed in either contravariant or covariant components into the
invariant distance between points. This is the rationale behind the statement
that the metric tensor “provides the geometry” of the space.
the role of the metric tensor in defining the length of a vector such as A
the role of the metric tensor in defining the angle between two vectors A and B
This explains why you’re likely to run into the statement that the metric tensor
“provides a dot product” for a space – if you know how to find the dot product,
you can define lengths and angles.
展开ds^2：
g21 = g 12 , g31 =g13 , g32 = g23
简记为：
recall：
that's why: g ij = ei ◦ ej
如何求matric tensor ：
(g$_{ij}$ g$^{ij}$ = I)
r , θ , φ -->x,y,z：gij，可以观察到一些特点。spherical polar coordinate axes, while curved, are orthogonal (that is, the lines of increasing r , θ , and φ intersect at right angles).
同时通过ds^2可以观察到距离是如何增加的。
这就是metric tensor的本质特性
For orthogonal coordinate systems, the square roots of the diagonal elements of the metric tensor
are called the “scale factors”
知道metric tensor 和 scale factors后，求grand， div，curl就很简单了(hi= sqrt(gii))
5.6 Index raising and lowering
gꜟⱼ Aʲ = Aꜟ
gⁱʲ Aⱼ= Aʲ
Christoffel symbols：
左边是求和约定。(
是scalar)
如果知道了matric tensor，就能很容易求得Christoffel symbols:(推导过程P150)
covariant derivative主要是为了把basis vector的变化也考虑进去。
考虑向量A= A¹e₁ + A² e₂ + A³ e₃，以及在
最后一行第二项里的偏导项就是“basis vector的变化率”，再乘上Aⁱ就是总的变化量。
上面的式子最终可化为：
covariant derivative的形式化定义：use a semicolon (;) in front of the index with respect to which the covariant derivative is being taken ( j in this case)
(右边是求和约定，展开∂A/∂xʲ
= ∑covariant derivative* basis vector
= ∑( ∂Ai/∂xʲ + Aᵏ * Гⁱₖⱼ )* basis vector )
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Most texts use one of two ways to derive the Riemann curvature tensor:
parallel transport or the commutator of the covariant derivative.
这个矩阵就是inertia tensor ，记为Iab。Iab tells you how much angular momentum in the a-direction
is produced by rotation about the b-axis.
x, v, a, ~ θ, ω, α(angular acceleration)
F, m, p, ~ τ(torque), I(moment of inertia), L(angular momentum)
spherical top
symmetric top
asymmetric top
Maxwell’s Equations:
The integral forms describe the behavior of electric and magnetic fields over surfaces or around paths,
The differential forms apply to specific locations.
a tensor version of the electromagnetic field equations and
a four-vector version of the Lorentz force law,
J = (cρ, Jx , Jy , Jz )
electromagnetic field tensor(若干版本之一。为什么会有多个版本？见本书网站http://www.danfleisch.com/sgvt/EMTensor.pdf):
Maxwell’s Equations can be expressed as:
同一个Fab，在坐标系A和A‘下对应着不同的值，关系是：
F’ = A Fab A⁻¹
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An object’s inertial mass determines its resistance to acceleration,
An object’s gravitational mass determines its response to a gravitational field.
The equality of these differently defined masses cannot be explained
by classical mechanics, and Einstein’s scientific instincts told him that the
resolution of this deficiency could be achieved by “an extension of the prin-
ciple of relativity to spaces of reference which are not in uniform motion
relative to one another.”
Riemann curvature tensor：
Ricci tensor:
Ricci scalar():
Einstein tensor:
Einstein’s field equation for GR:
Tμν :the energy-momentum tensor
Г: the “cosmological constant” introduced by Einstein to maintain a static Universe.
It is this equation
that gives rise to the first half of the concise statement of General Relativity:
“Matter tells spacetime how to curve, and spacetime tells matter how to move.”
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How represent vector and scalar in tensor symbol? Ai and A?
ˉ⁻⁰¹²³⁴⁵⁶⁷⁸⁹ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᴶᵏˡᵐⁿᵖʳˢᵗᵘᵛⱽʷˣʸᶻ
ˍ₋₁₂₃₄₅₆₇₈₉ₐₑₕꜟⱼₖₗₙₘₒₚₛₜₓₔᶿᶟᵉ
∂∑Г
αβγδεζηθικλμνξοπρςστυφχψω
