TongZhuo's Blog

二重积分的计算法 直角坐标 以这道题说明方法 先对Y积分，因为Y的范围一下就看出来了，而且对X的范围用y表示的时候也比较方便 ∫−12dy∫y2y+2dx\int_{-1}^{2}dy\int_{y^2}^{y+2}dx∫−12​dy∫y2y+2​dx 也可以换一下积分顺序，但是比较麻烦 有的原函数不能用初等函数表示，所以积分顺序是唯一的，弄错了就积不出来 利用对称性和奇偶性简化计算 积分区域关于x轴对称时，被积函数是关于y的奇函数可以抵消，偶函数翻倍 关于y轴对称时也一样 以a(x,y)=2x2ya(x,y)=2 x^2 ya(x,y)=2x2y的图像进行说明，不想死记公式，公式记多不仅痛苦还容易忘 积分区域关于x轴对称，关于y的奇函数，一半体积向上，一半向下，抵消了 极坐标 对x和y的积分变成了对θ和ρ的积分 判断θ的范围：以x轴的正方向为标准，从0开始，π2\frac{π}{2}2π​到y轴的正方向，−π2-\frac{π}{2}−2π​到y轴的负方向，看转多少范围可以把积分区域包起来，范围就是多少，转一圈才能把积分区域包起来的话，那就是0 ~...

释义列表 comedy - n. paraphrase：funny play or film synonym：humor, farce example：The comedy was so amusing that the audience laughed even when the actors forgot their lines. consumption - n. paraphrase：act of using up resources synonym：use, expenditure example：The government urged people to reduce energy consumption because power plants were overloaded. defeat - v./n. paraphrase：win against someone; failure synonym：beat, overcome example：If we defeat this team, we’ll advance to the...

type one：求极限 还是之前那种套路，和一元函数求极限没什么区别，不贴题了，过 type two：复合函数求偏导 还是链式法则的运用，哎呀，可烦这种套娃题型 第一种是这样，把函数都亮出来了，这个比较明晰，就是算起来可繁琐 (1) 设 z=ln⁡x2+y2z = \ln \sqrt{x^2 + y^2}z=lnx2+y2​，则 x∂z∂x+y∂z∂y=x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} =x∂x∂z​+y∂y∂z​=. (2) 设 z=arctan⁡x+y1−xyz = \arctan \frac{x + y}{1 - xy}z=arctan1−xyx+y​，则 ∂z∂x+∂z∂y=\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} =∂x∂z​+∂y∂z​=. 第二种是给了一个抽象函数，抽象函数里边有好几个参数，like this 设 z=xyf(x2+y2)z = xy f(x^2 +...

释义列表 cunning - adj./n. paraphrase：clever at deceiving others synonym：sly, crafty example：Although the fox was cunning, it couldn’t escape the trap that the farmer had set near the fence. burst - v./n. paraphrase：break open suddenly or explode synonym：explode, rupture example：The balloon burst when it touched the thorny bush, which startled the children playing nearby. superfluous - adj. paraphrase：more than needed; unnecessary synonym：excess, extra example：The teacher asked us to...

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Video Link 【Segment 1: Industry Dominance】 For years, Nvidia and Intel have dominated the semiconductor industry. Nvidia’s graphics processors have been the backbone of AI training, running everything from chatbots to self-driving cars. While Intel’s chips have powered devices from personal computers to massive data centers. But now, a new contender is emerging, and it’s shaking up the entire landscape: Cambricon Technologies, a Beijing-based chip designer that’s rapidly gaining...

为什么特解的差是齐次方程的解？ 1. 理论基础 对于 线性非齐次微分方程： y′+a(x)y=b(x)（非齐次方程）y' + a(x)y = b(x) \quad \text{（非齐次方程）} y′+a(x)y=b(x)（非齐次方程） 若已知两个特解 y1y_1y1​ 和 y2y_2y2​，则它们的差 y0=y2−y1y_0 = y_2 - y_1y0​=y2​−y1​ 必为对应的 齐次方程： y′+a(x)y=0（齐次方程）y' + a(x)y = 0 \quad \text{（齐次方程）} y′+a(x)y=0（齐次方程） 的解。 2. 数学推导 • 代入 y1y_1y1​ 和 y2y_2y2​ 到非齐次方程： {y1′+a(x)y1=b(x)y2′+a(x)y2=b(x)\begin{cases} y_1' + a(x)y_1 = b(x) \\ y_2' + a(x)y_2 = b(x) \end{cases} {y1′​+a(x)y1​=b(x)y2′​+a(x)y2​=b(x)​ •...

单词释义 petition - n./v. paraphrase：formal request signed by many people synonym：appeal, plea example：They started a petition to save the park, which gathered over 1,000 signatures within a week. appropriate - adj./v. paraphrase：① (adj.) suitable; ② (v.) take something without permission synonym：① fitting, proper; ② seize, take example：The teacher chose an appropriate book for the class, but some students found it too easy. harvest - n./v. paraphrase：gather crops synonym：collect,...

type one：几何关系列出等式解方程 第一道： 曲线 y=f(x) 上点 P(x,y) 处的切线的斜率等于该点坐标之积的两倍，且该曲线过点 (0,2), 则 f(x)=f(x) =f(x)=. 解：由题意得 y′=2xyy' = 2xyy′=2xy, 即 y′−2xy=0y' - 2xy = 0y′−2xy=0, 解得 y=Ce−∫(−2x)dx=Cex2y = C e^{-\int (-2x) \mathrm{d}x} = Ce^{x^2}y=Ce−∫(−2x)dx=Cex2, 因为该曲线过点 (0,2)(0,2)(0,2), 所以 C=2C = 2C=2, 故 f(x)=2ex2f(x) = 2e^{x^2}f(x)=2ex2. 第二道： 设位于第一象限的曲线 y=f(x)y = f(x)y=f(x) 上任意一点 P(x,y)P(x,y)P(x,y) 的切线在 xxx 轴上的截距等于该点法线在 yyy 轴上的截距的相反数，且曲线经过点 (1,0)(1,0)(1,0), 求该曲线. 解：设切线方程为 Y−y=y′(X−x)Y - y =...

释义列表 index - n. paraphrase：a list of topics or names in a book, often at the end synonym：list, guide example：She checked the index of the textbook to find the chapter about climate change, but it wasn’t listed there. occupation - n. paraphrase：a job or activity you do regularly synonym：job, profession example：His occupation as a teacher kept him busy, though he sometimes missed working outdoors. implication - n. paraphrase：a possible result or meaning of...