变积系数

前述内容应该参见几何变换和射影几何（一）。
现补充下述内容：
平面上的一个仿射点变换σ\sigma引起了平面的一个向量变换σ¯\bar \sigma
（指平面上所有向量组成的集合S¯\bar S的变换）
p1′=Ap1+A0,p2′=Ap2+A0p_1'=Ap_1+A_0,p_2'=Ap_2+A_0
(p1′−p2′)=A(p1−p2)(p_1'-p_2')=A(p_1-p_2)
这说明P′Q′⃗\vec{P'Q'}的坐标只跟σ\sigma和向量m有关，而与m的有向线段中的起点的选择无关，因此可以规定集合S¯\bar S里的一个对应法则σ¯\bar \sigma,它把m对应到P′Q′⃗\vec{P'Q'},即：
σ¯(m)=σ¯(PQ⃗)=P′Q′⃗\bar\sigma(\mathbf{m})=\bar\sigma(\vec{PQ})=\vec{P'Q'}
由上述讨论知σ¯\bar\sigma是S¯\bar S的一个变换。

仿射变换的变积系数

设在平面上规定了一个定向，它用单位法向量ee代表，用(a,b)(a,b)表示以a，b为邻边的并且边界的环行方向为a到b的旋转方向的定向平行四边形的定向面积，易知：
a×b=(a,b)ea\times b=(a,b)e
定理 设仿射变换τ\tau在仿射标架I[O;e1,e2]I[O;e_1,e_2]中的公式为
(x′y′)=(a11a12a21a22)(xy)+(a1a2)\begin{pmatrix} x' \\ y' \end{pmatrix}= \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}+ \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}
对于任意不共线向量a，b设τ¯(a)=a′,τ¯(b)=b′\bar\tau(a)=a',\quad\bar\tau(b)=b',则有
(a′,b′)(a,b)=∣a11a12a21a22∣\frac{(a',b')}{(a,b)}=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}
proof.proof.
由于τ\tau是仿射变换，因此它把仿射标架I变成了仿射标架II，τ¯\bar\tau在I中的公式可算出e1′e_1'的I坐标是(a11,a21)′(a_{11},a_{21})',e2′e_2'的I坐标是(a12,a22(a_{12},a_{22}
设a,b的I坐标分别为(u1,v1),(u2,v2)(u_1,v_1),(u_2,v_2),则a′,b′a',b'的II坐标分别为(u1,v1),(u2,v2)(u_1,v_1),(u_2,v_2)。在坐标系I中计算得：
a×b=(u1e1+v1e2)×(u2e1+v2e2)=∣u1u2v1v2∣(e1×e2)\begin{aligned} a\times b=&(u_1e_1+v_1e_2)\times(u_2e_1+v_2e_2)\\ =&\begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix}(e_1\times e_2) \end{aligned}
因为a×b=(a,b)e,e1×e2=(e1,e2)ea\times b=(a,b)e,e_1\times e_2=(e_1,e_2)e所以
(a,b)=∣u1u2v1v2∣(e1,e2)(a,b)=\begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix}(e_1,e_2)
注意到此公式对平面上的任意向量a，b均成立，特别地，对于e1′,e2′e_1',e_2'有
(e1′,e2′)=∣a11a12a21a22∣(e1,e2)(e_1',e_2')=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}(e_1,e_2)
同理，在II中计算得：
(a′,b′)=∣u1u2v1v2∣(e1′,e2′)(a',b')=\begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix}(e_1',e_2')
因此
(a′,b′)(a,b)=(e1′,e2′)(e1,e2)=∣a11a12a21a22∣\frac{(a',b')}{(a,b)}=\frac{(e_1',e_2')}{(e_1,e_2)}=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}
易知如果仿射变换τ\tau在仿射坐标系I中的公式的系数矩阵为A，那么τ\tau在仿射坐标系II中的公式的系数矩阵为H−1AHH^{-1}AH,其中H是I到II的过渡矩阵。因为
∣H−1AH∣=∣A∣|H^{-1}AH|=|A|
所以仿射变换tautau的公式中的系数矩阵的行列式与仿射标架的选择无关。

仿射变换τ\tau的公式中的系数矩阵的行列式称为τ\tau的行列式记作dτd_\tau,仿射变换τ\tau按同一个比值∣dτ∣|d_\tau|来改变平面上所有(有面积)的图形的面积，因此把∣dτ∣|d_\tau|称为仿射变换τ\tau的变积系数。

一般情形下的变积系数

二维情形

设
{x=ρcosθy=ρsinθ\begin{cases} x=\rho\cos\theta \\ y=\rho\sin\theta \end{cases}
我们把上式看做由直角坐标平面ρOθ\rho O\theta到直角坐标平面xOyxOy的一种变换，即对于ρOθ\rho O\theta上的一点M′(ρ,θ)M'(\rho,\theta)通过变换变成了xOyxOy平面上的一点M(x,y)M(x,y),在两个平面各自限定的范围内，这种变换还是一一映射，

定理
T:x=x(u,v),y=y(u,v)T:x=x(u,v),y=y(u,v)
将平面uOvuOv上的闭区域D'变为xOy上的D，且满足

  1. x(u,v),y(u,v)x(u,v),y(u,v)在D′D'上具有一阶连续偏导数；
  2. 在D'上的雅各比矩阵J(u,v)=∂(x,y)∂(u,v)≠0J(u,v)=\frac{\partial(x,y)}{\partial(u,v)}\neq 0
  3. 变换T:D′→DT:D'\to D是一对一的，有
    dxdy=∣J(u,v)∣dudvdxdy=|J(u,v)|dudv

一般情形

J=[∂f∂x1⋯∂f∂xn]=[∂f1∂x1⋯∂f1∂xn⋮⋱⋮∂fm∂x1⋯∂fm∂xn]\mathbf J = \begin{bmatrix} \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix}

The Jacobian generalizes the gradient of a scalar (mathematics)|scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a scalar-valued multivariate function is the gradient and that of a scalar-valued function of single variable is simply its derivative. The Jacobian can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that a transformation imposes locally. For example, if (x′,y′)=f(x,y)(x',y')=f(x,y)is used to transform an image, the Jacobian Jf(x,y)J_f(x,y) describes how the image in the neighborhood of (x,y)(x,y) is transformed.

If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function does not need to be differentiable for the Jacobian to be defined, since only the partial derivatives are required to exist.

If p is a point in RnR^n and f is derivative at p, then its derivative is given by Jf(p)J_f(p). In this case, the linear map described by Jf(p)J_f(p) is the best linear approximation off near the point p, in the sense that

f(x)=f(p)+Jf(p)(x−p)+o(∥x−p∥)\mathbf f(\mathbf x) = \mathbf f(\mathbf p) + \mathbf J_{\mathbf f}(\mathbf p)(\mathbf x - \mathbf p) + o(\|\mathbf x - \mathbf p\|)

for x close top and where o is the Little-o_notation forx→px\to p and ∣∣x−p∣∣||x-p|| is the Euclidean distance between x andp.
Compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order:

f(x)=f(p)+f′(p)(x−p)+o(x−p).f(x) = f(p) + f'(p) (x - p) + o(x - p) .

In a sense, both the gradient and Jacobian are "derivative"—the former the first derivative of a ''scalar function'' of several variables, the latter the first derivative of a ''vector function'' of several variables.

The Jacobian of the gradient of a scalar function of several variables has a special name: the Hessian matrix, which in a sense is the "second derivative" of the function in question.

参考资料

  1. 解析几何 丘维声
  2. Jacobi Matirx,Wikipedia
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