Jacobi matrix雅克比矩阵_push forward数学中的含义-优快云博客

本文深入探讨了向量微积分中的推前运算概念及其与雅克比矩阵的关系，解释了如何将涉及向量的导数简化为向量操作，并详细介绍了推前运算在向量函数求导过程中的应用。

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cite from wikipedia

Vector calculus

Vector-by-vector

Each of the previous two cases can be considered as an application of the derivative of a vector with respect to a vector, using a vector of size one appropriately. Similarly we will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way.

The derivative of a vector function (a vector whose components are functions) $\mathbf{y} =\begin{bmatrix}y_1 \\y_2 \\\vdots \\y_m \\\end{bmatrix}$ , of an independent vector $\mathbf{x} =\begin{bmatrix}x_1 \\x_2 \\\vdots \\x_n \\\end{bmatrix}$ , is written (in numerator layout notation) as

$\frac{\partial \mathbf{y}}{\partial \mathbf{x}} =\begin{bmatrix}\frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n}\\\frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n}\\\vdots & \vdots & \ddots & \vdots\\\frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n}\\\end{bmatrix}.$

In vector calculus, the derivative of a vector function y with respect to a vector x whose components represent a space is known as the pushforward or differential, or the Jacobian matrix.

The pushforward along a vector function f with respect to vector v in R^m is given by $d\,\mathbf{f}(\mathbf{v}) = \frac{\partial \mathbf{f}}{\partial \mathbf{x}} \mathbf{v}.$