<<Vector Calculus>>笔记

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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http://book.douban.com/annotation/36251494/

<<Vector Calculus>>
by Paul C, Matthews

P4

Since the quantity of |b|*cosθ represents the component of the vector b in thedirection of the vector a, the scalar a * b can be thought of as the magnitudeof 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 Falong a curve only depends on the endpoints of the curve, not on the pathtaken by the curve

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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

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 hasthe 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.