Vector Calculus(向量微积分)

Vector Calcuclus

Vectors and Parametric Curves

Basic

Definition (Scalar, Point, Bi-point, Vector)

Scalar
A scalar $α ∈ \R$ is simply a real number.
 
Point, Bi-point
A point $r ∈ \R^2$ is an ordered pair of real numbers, $r = (x, y)$ with $x ∈ \R$ and $y ∈ \R$. Here the first coordinate x stipulates the location on the horizontal axis and the second coordinate y stipulates the location on the vertical axis. Given two points r and r′ in $\R^2$ the directed line segment with departure point r and arrival point r′ is called the bi-point r, r′ and is denoted by [r,r′]. We say that r is the tail of the bi-point [r, r′] and that r′ is its head. The Euclidean length or norm of bi-point [a, b] is simply the distance between a and b and it is denoted by $||[a,b]|| = \sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}$
 
Vector
A vector $\vec a ∈ \R^2$ is a codification of movement of a bi-point: given the bi-point [r, r′], we associate to it the vector $\overrightarrow {rr^′} = \begin{bmatrix}x' - x \\ y′ - y\end{bmatrix}$ stipulating a movement of $x' - x$ units from (x, y) in the horizontal axis and of $y' - y$ units from the current position in the vertical axis.The zero vector $\vec 0 = \begin{bmatrix}0 \\ 0\end{bmatrix}$ indicates no movement in either direction.

Let $\vec u \ne \vec 0$. Put $\R\vec u = \{\lambda\vec u: \lambda \in \R\}$ and let $a \in \R^2$, the affine line with direction vector $\vec u = \begin{bmatrix}u_1 \\ u_2\end{bmatrix}$ and passing through a is the set of points on the plane.

$a + \R\vec u = \{\begin{pmatrix} x\\y \end{pmatrix} \in \R^2: x = a_1 + tu_1, y = a_2 + tu_2, t \in \R\}$

that is, the affine line is the Cartesian line with slope $\frac{u_2}{u_1}$, Conversely, if $y = mx + k$ is the equation of a Cartesian line, then

$\begin{pmatrix} x\\y \end{pmatrix} = \begin{bmatrix}1\\m\end{bmatrix}t + \begin{pmatrix} 0\\k \end{pmatrix}$

that is, every Cartesian line is also an affine line and one may take the vector $\begin{bmatrix}1\\m\end{bmatrix}$ as its direction vector.

Let $\vec x \in \R^2$ and $\vec y \in \R^2$. Their scalar product (dot product, inner product) is defined and denoted by $\vec x \cdot \vec y = x_1y_1 + x_2y_2$

Consider now two arbitrary vectors in $\R^2 , \vec x, \vec y$ , say. Under which conditions can we write an arbitrary vector $\vec v$ on the plane as a linear combination of x and y , that is, when can we find scalars a, b such that $\vec v = a \vec x + b \vec y$ ?

$\vec v = a \vec x + b \vec y \iff v_1 = ax_1 + by_1, v_2 = ax_2 + by_2\iff a = \frac{v_1y_2 - v_2y_1}{x_1y_2 - x_2y_1} , b = \frac{x_1v_2 - x_2v_1}{x_1y_2 - x_2y_1}$
The above expressions for a and b make sense only if $x_1y_2 \ne x_2y_1$.

So given two vectors in $\R^2 , \vec x, \vec y$ , an arbitrary vector $\vec v$ can be written as the linear combination $\vec v = a \vec x + b \vec y, a \in \R, b \in \R$ if and only if $\vec x$ is not parallel to $\vec y$.

Geometric Transformations in two dimensions

We now are interested in the following fundamental functions of sets on the plane: translations, scalings (stretching or shrinking) reflexions about the axes, and rotations about the origin.

Translate
A function $T_{\vec v}: \R^2 \to \R^2$ is said to be a translation if it is of the form $T_{\vec v}(x) = x + \vec v$, where $\vec v$ is a fixed vector on the plane.A translation simply shifts an object on the plane rigidly (that is, it does not distort it shape or re-orient it), to a copy of itself a given amount of units from where it was.
 
Scale
A function $S_{a,b}: \R^2 \to \R^2$ is said to be a scaling if it is of the form $S_{a,b}(r) = \begin{pmatrix}ax\\by\end{pmatrix}, a,b \in \R^+$
 
Reflexion
A function $R_H: \R^2 \to \R^2$ is said to be a reflexion about the y-axis or horizontal reflexion if it is of the form $R_H(r) = \begin{pmatrix}-x\\y\end{pmatrix}$
A function $R_V: \R^2 \to \R^2$ is said to be a reflexion about the x-axis or vertical reflexion if it is of the form $R_V(r) = \begin{pmatrix}x\\-y\end{pmatrix}$
A function $R_O: \R^2 \to \R^2$ is said to be a reflexion about the orgin if it is of the form $R_O(r) = \begin{pmatrix}x\\--y\end{pmatrix}$
 
Rotation
A function $R_\theta: \R^2 \to \R^2$ is said to be a levogyrate rotation about the origin by the angle θ measured from the positive x-axis if $R_\theta(r) = \begin{pmatrix}xcos\theta - ysin\theta \\ xsin\theta + ycos\theta \end{pmatrix}$
 
linear transformation
A function $L: \R^2 \to \R^2$ is said to be a linear transformation from $\R^2$ to $\R^2$ if for all points a, b on the plane and every scalar λ, it is verified that $L(a+b) = L(a) + L(b), L(\lambda a) = \lambda L(b)$
 
affine transformation
A function $A : \R^2 \to \R^2$ is said to be an affine transformation from $\R^2$ to $\R^2$ if there exists a linear transformation $L: \R^2 \to \R^2$ and a fixed vector $\vec v ∈ \R^2$ such that for all points $x ∈ \R^2$ it is verified that $A(x) = L(x) + \vec v$

Let $L : \R^2 \to \R^2$ be a linear transformation. The matrix $A_L$ associated to L is the 2 × 2, (2rows, 2columns) array whose columns are in this order $L\begin{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}\end{pmatrix}$ and $L\begin{pmatrix}\begin{pmatrix}0\\1\end{pmatrix}\end{pmatrix}$

Determinants in two dimensions

The determinant of the 2 × 2 matrix $\begin{bmatrix}a & c\\b & d\end{bmatrix}$ is $det\begin{bmatrix}a & c\\b & d\end{bmatrix} = ad - bc$.

Consider now a simple quadrilateral with vertices $r_1 = (x_1, y_1), r_2 = (x_2, y_2), r_3 = (x_3, y_3), r_4 = (x_4, y_4)$, listed in counterclockwise order. This quadrilateral is spanned by the vectors

$\overrightarrow{r_1r_2} = \begin{bmatrix}x_2 -x_1\\y_2-y_1\end{bmatrix}, \overrightarrow{r_1r_4} = \begin{bmatrix}x_4 -x_1\\y_4-y_1\end{bmatrix}$

and hence, its area is given by

$A = det\begin{bmatrix}x_2 - x_1 & x_4 - x_1 \\ y_2 - y_1 & y_4 - y_1\end{bmatrix} = D(\vec r_2 - \vec r_1, \vec r_4 - \vec r_1)$

We conclude that the area of a quadrilateral with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$, listed in counter clockwise order is

$\frac{1}{2}(det\begin{pmatrix}x_1 & x_2\\y_1 & y_2\end{pmatrix} + det\begin{pmatrix}x_2 & x_3\\y_2 & y_3\end{pmatrix} + det\begin{pmatrix}x_3 & x_4\\y_3 & y_4\end{pmatrix} + det\begin{pmatrix}x_4 & x_1\\y_4 & y_1\end{pmatrix} )$

In general, we have the following theorem.

Let $(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$ be the vertices of a simple (non- crossing) polygon, listed in counterclockwise order. Then its area is given by

$\frac{1}{2}(det\begin{pmatrix}x_1 & x_2\\y_1 & y_2\end{pmatrix} + det\begin{pmatrix}x_2 & x_3\\y_2 & y_3\end{pmatrix} + ... + det\begin{pmatrix}x_{n-1} & x_{n}\\y_{n-1} & y_n\end{pmatrix} + det\begin{pmatrix}x_n & x_1\\y_n & y_1\end{pmatrix})$

Parametric Curves on the Plane

Let $[a; b] ⊆ \R$. A parametric curve representation r of a curve Γ is a function $r : [a; b] \to \R$ , with

$r(t) = \begin{pmatrix}x(t)\\y(t)\end{pmatrix}$

and such that $r([a; b]) = Γ$. r(a) is the initial point of the curve and r(b) its terminal point.

As the determinants in two dimensions corrensponding the area. $S_\Delta =\frac{1}{2} det\begin{bmatrix}x & x + \Delta x \\ y & y + \Delta y\end{bmatrix} = \frac{1}{2} (x\Delta y - y\Delta x)$
So for the parametric curves which can be seen as $N(N\to\infty)$ points. So we have

$S_{pc} = \frac{1}{2}\oint_\Gamma(xdy - ydx)$

Vectors in Space

The 3-dimensional Cartesian Space is defined and denoted by $\R^3 = \{r = (x,y,z): x\in \R, y \in \R, z \in R\}$
The dot product of two vectors $\vec a$ and $\vec b$ in $\R^3$ is $\vec a \cdot \vec b = a_1b_1 + a_2b_2 + a_3b_3$

Let $\vec u$ and $\vec v$ be linearly independent vectors. The parametric equation of a plane containing the point a, and parallel to the vectors $u$ and $v$ is given by

$\vec r - \vec a = p\vec u + q\vec v$

in the other way, the equation of the plane in space can be written in the form $ax + by + cz = d$, which is the product of any $\vec v = (x + d_1,y+d_2,z+d_3)$ and fixed $\vec p = (a, b, c)$ with condition $ad_1 + bd_2 + cd_3 = 0$

Cross Product

We now define the standard cross product in $\R^3$ as a product satisfying the following properties.

Let $\vec x, \vec y, \vec z$ be vectors in $\R^3$ ,and let $α∈R$ be a scalar. The cross product $×$ is a closed binary operation satisfying

Anti-commutativity: $\vec x \times \vec y = -(\vec y \times \vec x)$
Bilinearity: $(\vec x + \vec z) \times \vec y = \vec x \times \vec y + \vec z \times \vec y$
Scalar homogeneity: $(a\vec x) \times \vec y = \vec x \times (a\vec y) = a(\vec x \times \vec y)$
Zeror Rule: $\vec x \times \vec x = \vec 0$
Right-hand Rule: $\vec i \times \vec j = \vec k, \vec j \times \vec k = \vec i, \vec k \times \vec i = \vec j$

Let $\vec x = \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$ and $\vec y = \begin{bmatrix}y_1\\y_2\\y_3\end{bmatrix}$ be vectors in $\R^3$ . Then

$\vec x\times \vec y = (x_2y_3 - x_3y_2)\vec i - (x_1y_3 - x_3y_1)\vec j + (x_1y_2 - x_2y_1)\vec k$

With the definition, it’s easy to get that

1: $\vec x \perp (\vec x + \vec y) \space and \space \vec y \perp (\vec x + \vec y)$
2: $\vec a \times (\vec b + \vec c) = (\vec a \cdot \vec c)\vec b - (\vec a \cdot \vec b)\vec c$
3: denote the convex angle between two vector $\vec x , \vec y$ is $\theta$, then $||\vec x \times \vec y|| = ||\vec x|||\vec y||sin\theta$

corolary: Two non-zero vectors $\vec x, \vec y$ , satisfy $\vec x × \vec y = \vec 0$ if and only if they are parallel.

Let $\vec a , \vec b , \vec c$ be linearly independent vectors in $\R^3$ . The signed volume of the parallelepiped spanned by them is $(\vec a × \vec b)• \vec c$.

Matrices in three dimensions

We will briefly introduce 3 × 3 matrices. Most of the material will flow like that for 2 × 2 matrices.

A linear transformation $T : \R^3 \to \R^3$ is a function such that $T(a + b) = T(a) + T(b), T(λa) = λT(a)$, for all points a, b in $\R^3$ and all scalars λ. Such a linear transformation has a 3×3 matrix representation whose columns are the vectors $T (i), T (j), T (k)$.

Determinants in three dimensions

Since thanks to Theorem, the signed volume of the parallelepiped spanned by them is $(\vec a × \vec b)• \vec c$, we define the determinant of A, det A, to be

$D(\vec a, \vec b, \vec c) = det\begin{bmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} = \vec a \cdot (\vec b \times \vec c)$

Spherical Trigonometry

Given a point (x, y, z) in Cartesian coordinates, its spherical coordinates are given by

$x = ρcosθsinφ, y = ρsinθsinφ,z = ρcosφ$

Here φ is the polar angle, measured from the positive z-axis, and θ is the azimuthal angle, measured from the positive x-axis. By convention, 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π.

Spherical coordinates are extremely useful when considering regions which are symmetric about a point.

Canonical Surfaces

In this section we consider various surfaces that we shall periodically encounter in subsequent sec- tions. Just like in one-variable Calculus it is important to identify the equation and the shape of a line, a parabola, a circle, etc., it will become important for us to be able to identify certain families of often-occurring surfaces. We shall explore both their Cartesian and their parametric form. We remark that in order to parametrise curves (“one-dimensional entities”) we needed one parameter, and that in order to parametrise surfaces we shall need to parameters.

Parametric Curves in Space

Let $[a; b] ⊆ \R$. A parametric curve representation r of a curve $Γ$ is a function $r : [a; b] \to \R^3$

$r(t) = \begin{pmatrix}x(t) \\ y(t) \\z(t) \end{pmatrix}$

and such that $r([a; b]) = Γ$. r(a) is the initial point of the curve and r(b) its terminal point. A curve is closed if its initial point and its final point coincide. The trace of the curve r is the set of all images of r, that is, $Γ$. The length of the curve is
$\int_\Gamma||d\vec r||$

Multidimensional Vectors

We briefly describe space in n-dimensions. The ideas expounded earlier about the plane and space carry almost without change.

$\R^n$ is the n-dimensional space, the collection $\R^n = \{ \begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}: x_k \in \R \}$

Given vectors $\vec a , \vec b$ of $\R^n$ , their dot product is $\displaystyle \vec a \cdot \vec b = \sum_{k=1}^n a_kb_k$

Cauchy-Bunyakovsky-Schwarz Inequality: Let $\vec x$ and $\vec y$ be any two vectors in $\R^n$, then we have $|\vec x \cdot \vec y | ≤ ||\vec x||\vec y|| .$.
The form of the Cauchy-Bunyakovsky-Schwarz most useful to us will be $\displaystyle |\sum_{k=1}^nx_ky_k|\le(\sum_{k=1}^nx_k^2)^{\frac{1}{2}}(\sum_{k=1}^ny_k^2)^{\frac{1}{2}}$

Differentiation

Multivariable Functions

Let $A ⊆ \R^n$ . For most of this course, our concern will be functions of the form $f: A\to\R^m$
If $m=1$,we say that $f$ is a scalar field. If $m≥2$,we say that $f$ is a vector field.

Definition of the Derivative

Let $A⊆\R^n$.A function $f:A\to\R^m$ is said to be differentiable at $a∈A$ if there is a linear transformation, called the derivative of f at a, $D_a(f): \R^n\to\R^m$ such that

$\displaystyle \lim_{x\to a} = \frac{||f(x) - f(a) - D_a(f)(x-a)||}{||x - a||} = 0$

The Jacobi Matrix

We now establish a way which simplifies the process of finding the derivative of a function at a given point.
Let $A⊆\R^n ,f :A\to \R^m$ ,and put

$\displaystyle f(x) = \begin{bmatrix}f_1(x_1, x_2, \cdots ,x_n) \\ f_2(x_1, x_2, \cdots ,x_n) \\ \vdots \\ f_m(x_1, x_2, \cdots ,x_n) \\ \end{bmatrix}$

Here $f_i: \R^n \to \R$. The partial derivative $\frac{\partial f_i}{\partial x_j}(x)$ is defined as

$\displaystyle \frac{\partial f_i}{\partial x_j}(x) = \lim_{h\to 0}\frac{f_i(x_1, x_2, \cdots, x_j + h, \cdots, x_n) - f_i(x_1, x_2, \cdots, x_j, \cdots, x_n) }{h}$

whenever this limit exists.

To find partial derivatives with respect to the j-th variable, we simply keep the other variables fixed and differentiate with respect to the j-th variable.

Then each partial derivative $\frac{\partial f_i}{\partial x_j}(x)$ exists, and the matrix representation of $D_x(f)$ with respect to the standard bases of $\R^n$ and $\R^m$ is the Jacobi matrix

$\displaystyle f'(x) = \begin{bmatrix} \frac{\partial f_1}{\partial x_1}(x) & \frac{\partial f_1}{\partial x_2}(x) & \cdots & \frac{\partial f_1}{\partial x_n}(x) \\ \frac{\partial f_2}{\partial x_1}(x) & \frac{\partial f_2}{\partial x_2}(x) & \cdots & \frac{\partial f_2}{\partial x_n}(x) \\ \vdots & \vdots & \vdots & \vdots \\ \frac{\partial f_n}{\partial x_1}(x) & \frac{\partial f_n}{\partial x_2}(x) & \cdots & \frac{\partial f_n}{\partial x_n}(x)\end{bmatrix}$

for eample, the function $f(x,y) = (xy + yz, log_exy)$, then jacobi matrix is $f'(x,y) = \begin{bmatrix}y & x+z & y \\ \frac{1}{x} & \frac{1}{y} & 0 \end{bmatrix}$

Gradients and Directional Derivatives

A function $f: \bold x \in \R^n \to f(\bold x) \in \R^m$ is called a vector field.
If $m = 1$, it is called a scalar field.

Definition

Let $f:\begin{matrix} \R^n & \to & \R \\ \bold x & \to & f(\bold x) \end{matrix}$ be a scalar field.
The gradient of $f$ is the vector defined and denoted by $\displaystyle \nabla f(\bold x) = \begin{bmatrix}\frac{\partial f}{\partial x_1}(\bold x) \\ \frac{\partial f}{\partial x_2}(\bold x) \\ \vdots \\ \frac{\partial f}{\partial x_n}(\bold x) \end{bmatrix}$

The gradient operator is the operator $\displaystyle \nabla = \begin{bmatrix}\frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \vdots \\ \frac{\partial}{\partial x_n}\end{bmatrix}$

Definition

Let $f:\begin{matrix} \R^n & \to & \R^n \\ \bold x & \to & f(\bold x) \end{matrix}$ be a vector field with $f(\bold x) = \begin{bmatrix}f_1(\bold x) \\ f_2(\bold x) \\ \vdots \\ f_n(\bold x)\end{bmatrix}$
The divergence of f is defined and denoted by $\displaystyle div f(\bold x) = \nabla \cdot f(\bold x) = \frac{\partial f_1}{\partial x_1}(\bold x) + \frac{\partial f_2}{\partial x_2}(\bold x) + \cdots + \frac{\partial f_n}{\partial x_n}(\bold x)$

Extrema (and Hessian matrix)

We now turn to the problem of finding maxima and minima for vector functions. As in the one-variable case, the derivative will provide us with information about the extrema, and the “second derivative” will provide us with information about the nature of these extreme points.

To define an analogue for the second derivative, let us consider the following. Let $A ⊂ \R^n$ and $f : A \to \R^m$ be differentiable on A. We know that for fixed $x0 ∈ A$, $D_{x_0}(f)$ is a linear transformation from $\R^n$ to $\R^m$. This means that we have a function

$\displaystyle T:\begin{matrix}A & \to & L(\R^n, \R^m) \\ x & \to & D_x(f)\end{matrix}$

where $L(\R^n, \R^m)$ denotes the space of linear transformations from $\R^n$ to $\R^m$. Hence, if we differentiate T at $x_0$ again, we obtain a linear transformation $D_{x_0}(T) = D_{x_0}(D_{x_0}(f)) = D_{x_0}^2(f)$ from $\R^n$ to $L(\R^n, \R^m)$. Hence, given $x_1 ∈ \R^n$, we have $D_{x_0}^2(f)(x_1) ∈ L (\R^n, \R^m)$. Again, this means that given $x_2 ∈ \R^n, D_{x_0}^2(f)(x_1))(x_2) ∈ \R^m $. Thus the function

$B_{x_0}:\begin{matrix}\R^n \times \R^n & \to & L(\R^n, \R^m) \\ (x_1, x_2) & \to & D_{x_0}^2(f)(x_1, x_2)\end{matrix}$

is well defined, and linear in each variable $x_1$ and $x_2$, that is, it is a bilinear function.

Theorem:

Let $A ⊆ \R^n$ be an open set, and $f : A \to \R$ be twice differentiable on A. Then the matrix of $D_{x}^2(f) : \R^n × \R^n \to \R$ with respect to the standard basis is given by the Hessian matrix:

$H_xf = \begin{bmatrix} \frac{\partial ^2f}{\partial x_1 \partial x_1}(x) & \frac{\partial ^2f}{\partial x_1 \partial x_2}(x) & \cdots & \frac{\partial ^2f}{\partial x_1 \partial x_n}(x) \\ \frac{\partial ^2f}{\partial x_2 \partial x_1}(x) & \frac{\partial ^2f}{\partial x_2 \partial x_2}(x) & \cdots & \frac{\partial ^2f}{\partial x_2 \partial x_n}(x) \\ \vdots & \vdots & \vdots & \vdots \\ \frac{\partial ^2f}{\partial x_n \partial x_1}(x) & \frac{\partial ^2f}{\partial x_n \partial x_2}(x) & \cdots & \frac{\partial ^2f}{\partial x_n \partial x_n}(x) \end{bmatrix}$

Lagrange Multipliers

Integration

[Todo]

Reference

Multivariable Geometry and Vector Calculus By David A. SANTOS
Vector Calculus By Susan Jane Colley