Cynosure对《Vector Calculus》的笔记(1)

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现在流行用Exterior Caculus, 所以个人觉得Matthews这本书有点过时了。

想学Vector Calculus的话，推荐《Vector Calculus, Linear Algebra, and Differential Forms》，网上有第一版的电子版。虽然出到了第五版，但貌似vector caculus 和differential forms的部分没有什么改动。所以个人觉得用第一版学习vector caculus足以。
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P4
Since the quantity of |b|*cosθ represents the component of the vector b in the
direction of the vector a, the scalar a * b can be thought of as the magnitude
of a multiplied by the component of b in the direction of a 

P7
the general form of the equation of a plane is: r * a = constant.

P11
          | e1    e2   e3  |
a x b=|  a1   a2    a3  |
         |  b1   b2     b3 |

v = Ω x r

P24
The equation of a line is: r = a + λu
The second equation of a line is: r x u = b = a x u



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1.4 Scalar triple product （[a, b, c]）
The dot and the cross can be interchanged:
[a, b, c]≡a * b x c = a x b * c

The vectors a, b and c can be permuted cyclically:
a * b x c = b * c x a = c * a x b

The scalar triple product can be written in the form of a determinant:
               | a1    a2      a3  |
a * b x c= |  b1   b2     b3  |
               |  c1   c2       c3 |
If any two of the vectors are equal, the scalar triple product is zero.


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1.5 Vector triple product   a x (b x c)

a x (b x c) =  (a * c)*b - (a * b)*c
(a x b) x c = -(b * c)*a + (c * a)*b
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1.6 Scalar fields and vector fields
A scalar or vector quantity is said be a field if it is a function of position.

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2.2.3 Conservative vector fields
A vector field F is said to be conservative if it has the property that the line integral of F around any closed curve C is zero:

An equivalent definition is that F is conservative if the line integral of F
along a curve only depends on the endpoints of the curve, not on the path
taken by the curve


2.3.2

3.1.2 Taylor series in more than one variable

3.2 Gradient of a scalar field




The symbol ∇ can be interpreted as a vector differential operator,

where the term operator means that ∇ only has a meaning when it acts on some other quantity.


Theorem 3.1
Suppose that a vector field F is related to a scalar field Φ by F = ∇Φ and ∇ exists everywhere in some region D. Then F is conservative within D.
Conversely, if F is conservative, then F can be written as the gradient of a scalar field, F = ∇Φ.



If a vector field F is conservative, the corresponding scalar field Φ which obeys F = ∇Φ is called the potential（势能） for F.
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3.3.2 Laplacian of a scalar field
3.3.2 Laplacian of a scalar field


4.3 The alternating tensor εijk

5.1.1 Conservation of mass for a fluid
5.1.1

6.1 Orthogonal curvilinear coordinates
P100
Suppose a transformation is carried out from a Cartesian coordinate system (x1, x2, x3) to another coordinate system (u1, u2, u3)

e1 =(∂x/∂u1) / h1,   h1 = | ∂x/∂u1 |
e2 =(∂x/∂u2) / h2,   h2 = | ∂x/∂u2 |
e3 =(∂x/∂u3) / h3,   h3 = | ∂x/∂u3 |

dS = h1 * h2          *   du1 * du2
dV = h1 * h2 * h3   *   du1 * du2 * du3
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相关内容在《微积分学教程（第三卷）》（by 菲赫金哥尔茨）里使用Jacobi式阐述的：
16章
 $4. 二重积分中的变量变换
603.平面区域的变换
604.例1）（极坐标的例子）
605.曲线坐标中面积的表示法
607.几何推演
609.二重积分中的变量变换

17章 曲面面积，曲面积分
619. 例2 （引入A，B，C）
626 曲面面积的存在及其计算
629 例14）球面极坐标的计算

18章 三重积分及多重积分
$3 三重积分中的变量变换
655. 空间的变换及曲线坐标
656 例1 圆柱坐标，例2球坐标
657 曲线坐标下的体积表示法 （得出曲面坐标下的体积元素）
659 几何推演
661 三重积分中的变量变换
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Summary of Chapter 6
The system (u1, u2, u3) is orthogonal if ei * ej = δij.




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7. Cartesian Tensors

7.1 Coordinate transformations

A matrix with this property, that its inverse is equal to its transpose, is said to be orthogonal。

So far we have only considered a two-dimensional rotation of coordinates. Consider now a general three-dimensional rotation. For a position vector x = x1e1 + x2e2 + x3e3, 
x' = e'i * x   (x在e'i上的投影)   = e'i * (e1*x1 + e2*x2 + e3*x3) = e'i * ei*xi
xi = Lji * x'j  ..........................(7.6)

7.2 Vectors and scalars
A quantity is a tensor if each of the free suffices transforms according to the rule (7.4).

Lij * Lkj = δik



7.3.3 Isotropic tensors
The two tensors δij and εijk have a special property. Their components are the same in all coordinate systems. A tensor with this property is said to be isotropic.


7.4 Physical examples of tensors
7.4.1 Ohm's law
This is why δik is said to be an isotropic tensor: it represents the relationship between two vectors that are always parallel, regardless of their direction.

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8 Applications of Vector Calculus

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8.5 Fluid mechanics

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Example 8.12
Choosing the x-axis to be parallel to the channel walls, the velocity u has
the form u = (u, 0, 0). As the fluid is incompressible(所有点的速度(沿x轴)相同), ∇u = 0, so ∂u/∂x = 0.