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‎blog/2025-04/_partials/calculus.mdx‎

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@@ -109,52 +109,52 @@ $$
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A_{11} & \cdots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{m1} & \cdots & A_{mn} \
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\end{bmatrix}$"}：
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<Collapse label="$$
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\nabla_x (A x) = A^T
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$$">
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$$
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\begin{split}
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&\ \nabla_x (A x) = \nabla_x (A_1 x_1 + A_2 x_2 + \cdots + A_n x_n) \\
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=&\ \nabla_x ([A_{11} \cdots A_{1m}] x_1 + [A_{21} \cdots A_{2m}] x_2 + \cdots + [A_{n1} \cdots A_{nm}] x_n) \\
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=&\ \nabla_x (A_1 x_1 + A_2 x_2 + \cdots + A_n x_n) \\
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=&\ \nabla_x (A_1 x_1) + \nabla_x (A_2 x_2) + \cdots + \nabla_x (A_n x_n) \\
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=&\ \begin{bmatrix}
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\frac{\partial}{\partial x_1} (A_1 x_1) , \frac{\partial}{\partial x_2} (A_2 x_2) , \cdots , \frac{\partial}{\partial x_n} (A_n x_n)
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\end{bmatrix} \\
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=&\ \begin{bmatrix}
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A_1 , A_2 , \cdots , A_n
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\end{bmatrix} = A^T
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\end{split}
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$$
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</Collapse>
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<Collapse label="$$
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\nabla_x (A x) = A^T
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$$">
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$$
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\begin{split}
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&\ \nabla_x (A x) = \nabla_x (A_1 x_1 + A_2 x_2 + \cdots + A_n x_n) \\
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=&\ \nabla_x ([A_{11} \cdots A_{1m}] x_1 + [A_{21} \cdots A_{2m}] x_2 + \cdots + [A_{n1} \cdots A_{nm}] x_n) \\
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=&\ \nabla_x (A_1 x_1 + A_2 x_2 + \cdots + A_n x_n) \\
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=&\ \nabla_x (A_1 x_1) + \nabla_x (A_2 x_2) + \cdots + \nabla_x (A_n x_n) \\
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=&\ \begin{bmatrix}
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\frac{\partial}{\partial x_1} (A_1 x_1) , \frac{\partial}{\partial x_2} (A_2 x_2) , \cdots , \frac{\partial}{\partial x_n} (A_n x_n)
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\end{bmatrix} \\
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=&\ \begin{bmatrix}
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A_1 , A_2 , \cdots , A_n
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\end{bmatrix} = A^T
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\end{split}
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$$
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</Collapse>
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- 对于所有的 :ctip[$\mathbf{x} \in R^n$]{id="$\mathbf{x} = [x_1, x_2, \cdots, x_n]^T$"} \
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和 :ctip[$A \in R^{n \times m}$]{id="$\mathbf{A} = \begin{bmatrix} \
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A_{11} & \cdots & A_{1m} \\ \vdots & \ddots & \vdots \\ A_{n1} & \cdots & A_{nm} \
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\end{bmatrix}$"}：
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<Collapse label="$$
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\nabla_x (x^T A) = A
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$$">
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$$
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\begin{split}
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&\ \nabla_x (x^T A) = \nabla_x (x_1 A_1 + x_2 A_2 + \cdots + x_n A_n) \\
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=&\ \nabla_x (x_1 [A_{11} \cdots A_{1m}] + x_2 [A_{21} \cdots A_{2m}] + \cdots + x_n [A_{n1} \cdots A_{nm}]) \\
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=&\ \nabla_x (x_1 A_1 + x_2 A_2 + \cdots + x_n A_n) \\
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=&\ \nabla_x (x_1 A_1) + \nabla_x (x_2 A_2) + \cdots + \nabla_x (x_n A_n) \\
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=&\ \begin{bmatrix} \
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\frac{\partial}{\partial x_1} (x_1 A_1) , \frac{\partial}{\partial x_2} (x_2 A_2) , \cdots , \frac{\partial}{\partial x_n} (x_n A_n)
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\end{bmatrix} \\
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=&\ \begin{bmatrix}
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A_1 , A_2 , \cdots , A_n
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\end{bmatrix} = A
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\end{split}
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$$
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</Collapse>
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<Collapse label="$$
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\nabla_x (x^T A) = A
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$$">
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$$
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\begin{split}
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&\ \nabla_x (x^T A) = \nabla_x (x_1 A_1 + x_2 A_2 + \cdots + x_n A_n) \\
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=&\ \nabla_x (x_1 [A_{11} \cdots A_{1m}] + x_2 [A_{21} \cdots A_{2m}] + \cdots + x_n [A_{n1} \cdots A_{nm}]) \\
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=&\ \nabla_x (x_1 A_1 + x_2 A_2 + \cdots + x_n A_n) \\
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=&\ \nabla_x (x_1 A_1) + \nabla_x (x_2 A_2) + \cdots + \nabla_x (x_n A_n) \\
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=&\ \begin{bmatrix} \
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\frac{\partial}{\partial x_1} (x_1 A_1) , \frac{\partial}{\partial x_2} (x_2 A_2) , \cdots , \frac{\partial}{\partial x_n} (x_n A_n)
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\end{bmatrix} \\
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=&\ \begin{bmatrix}
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A_1 , A_2 , \cdots , A_n
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\end{bmatrix} = A
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\end{split}
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$$
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</Collapse>
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- **二次型**（二次型是二次函数在向量空间中的推广）：
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@@ -163,23 +163,23 @@ A_{11} & \cdots & A_{1m} \\ \vdots & \ddots & \vdots \\ A_{n1} & \cdots & A_{nm}
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A_{11} & \cdots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{n1} & \cdots & A_{nn} \
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\end{bmatrix}$"}：
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<Collapse label="$$
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\nabla_x x^T A x = (A + A^T) x
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$$">
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<Collapse label="$$
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\nabla_x x^T A x = (A + A^T) x
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$$">
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$$
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\begin{split}
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&\ \nabla_x x^T A x = \nabla_x \sum_{i=1}^n \sum_{j=1}^n x_i A_{ij} x_j \\
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=&\ \frac{\partial}{\partial x_k} \sum_{i=1}^n \sum_{j=1}^n x_i A_{ij} x_j + \frac{\partial}{\partial x_k} \sum_{i=1}^n \sum_{j=1}^n x_j A_{ji} x_i \\
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=&\ \sum_{i=1}^n \sum_{j=1}^n A_{ij} x_j + \sum_{i=1}^n \sum_{j=1}^n A_{ji} x_i \\
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=&\ \sum_{j=1}^n A_{kj} x_j + \sum_{i=1}^n A_{ik} x_i \text{（$i, j = k$ 时，$A_{kj} x_j, A_{ik} x_i$ 分别存在一项 $A_{kk} x_k$）} \\
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=&\ \sum_{i=1}^n (\sum_{j=1}^n A_{ij} x_j) \cdot e_i + \sum_{j=1}^n (\sum_{i=1}^n A_{ji} x_i) \cdot e_j \\
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=&\ \sum_{i=1}^n (\mathbf{A} \mathbf{x})_i \cdot e_i + \sum_{j=1}^n (\mathbf{A^T} \mathbf{x})_j \cdot e_j \\ \
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=&\ (A + A^T) x
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\end{split}
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$$
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$$
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\begin{split}
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&\ \nabla_x x^T A x = \nabla_x \sum_{i=1}^n \sum_{j=1}^n x_i A_{ij} x_j \\
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=&\ \frac{\partial}{\partial x_k} \sum_{i=1}^n \sum_{j=1}^n x_i A_{ij} x_j + \frac{\partial}{\partial x_k} \sum_{i=1}^n \sum_{j=1}^n x_j A_{ji} x_i \\
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=&\ \sum_{i=1}^n \sum_{j=1}^n A_{ij} x_j + \sum_{i=1}^n \sum_{j=1}^n A_{ji} x_i \\
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=&\ \sum_{j=1}^n A_{kj} x_j + \sum_{i=1}^n A_{ik} x_i \text{（$i, j = k$ 时，$A_{kj} x_j, A_{ik} x_i$ 分别存在一项 $A_{kk} x_k$）} \\
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=&\ \sum_{i=1}^n (\sum_{j=1}^n A_{ij} x_j) \cdot e_i + \sum_{j=1}^n (\sum_{i=1}^n A_{ji} x_i) \cdot e_j \\
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=&\ \sum_{i=1}^n (\mathbf{A} \mathbf{x})_i \cdot e_i + \sum_{j=1}^n (\mathbf{A^T} \mathbf{x})_j \cdot e_j \\ \
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=&\ (A + A^T) x
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\end{split}
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$$
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</Collapse>
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</Collapse>
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- **范数**：
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@@ -195,19 +195,19 @@ A_{11} & \cdots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{n1} & \cdots & A_{nn}
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A_{11} & \cdots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{n1} & \cdots & A_{nn} \
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\end{bmatrix}$"}：
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<Collapse label="$$
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\nabla_x \|x\|_2^2 = \nabla_x (x^T x) = 2x
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$$">
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<Collapse label="$$
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\nabla_x \|x\|_2^2 = \nabla_x (x^T x) = 2x
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$$">
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$$
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\begin{split}
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&\ \nabla_x \|x\|_2 = \nabla_x (\sqrt{x^T x}) ^ 2 \\
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=&\ \nabla_x (x^T x) = \nabla_x (x_1^2 + x_2^2 + \cdots + x_n^2) \\
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=&\ 2x
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\end{split}
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$$
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$$
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\begin{split}
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&\ \nabla_x \|x\|_2 = \nabla_x (\sqrt{x^T x}) ^ 2 \\
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=&\ \nabla_x (x^T x) = \nabla_x (x_1^2 + x_2^2 + \cdots + x_n^2) \\
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=&\ 2x
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\end{split}
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$$
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</Collapse>
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</Collapse>
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:::nerd
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在深度学习中每层神经网络之间由 **_权重矩阵_（通常还会添加同维度的偏置向量）桥接不同纬度矩阵的计算**。随后再通过:term[激活函数]{./terms/dl#activation-function}将计算结果映射到非线性空间，是:term[神经元]{./terms/dl#neuron}的计算核心。

‎blog/2025-08/prompt-engineering.mdx‎

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<!--truncate-->
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```markmap
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## **Tree of Thoughts (ToT)框架**
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- 基本信息
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- 提供代码库（含所有提示）
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```
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## 提示词要素
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常规的提示词通常包含以下要素：

‎src/components/markdown/svgviewer/Viewer.tsx‎

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);
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};
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export default ({
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svg,
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text
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}: {
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svg: React.ReactNode;
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text: string;
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}) => {
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export default ({ svg, text }: { svg: React.ReactNode; text: string }) => {
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const [isFullscreen, setIsFullscreen] = useState<boolean>(false);
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return (
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<ConfigProvider

‎src/components/markmap/View.tsx‎

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const mm = Markmap.create(refSvg.current, {
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...defaultOptions,
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zoom: true,
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duration: 0,
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duration: 0
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});
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mm.setData(transformed.root).then(() => {

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