中英文对照_数学常用短语与从句

数学常用短语与从句数学常用短语与从句 Phrases or clauses frequently used in mathematics 常见的涉及运算的短语或句子常见的涉及运算的短语或句子 1 Differentiate both sides of the equation and we get Integrate both sides of the equation and we get 2 Differentiating both sides of the equation we get Integrating both sides of the equation we get 3 Add A to B and we have 其中 A B 为某些表达式 如不等式 等式 方程等 下同 4 Subtract B from A and we have 5 Multiplying each term of the equation by we obtain 6 Dividing the equation through by we have 7 A and B together give A and B together yield A and B together imply 8 Comparing A with B it is easy to see that 9 Substituting A into B we obtain 10 Eliminating the parameter t from A and B we have 11 By introducing a new variable we can then rewrite A as follows By introducing a new variable we can then rewrite A in the following form 12 By a simple calculation we obtain from A 定理证明过程中常见的短语和句子定理证明过程中常见的短语和句子 1 下面的句型可用来表达下面的句型可用来表达 根据什么即可得到什么根据什么即可得到什么 的意思的意思 According to definition it follows According to hypothesis it follows According to assumptions it follows According to theorem N it follows According to lemma A it follows According to corollary B it follows According to the remark it follows According to the fact that it follows 可以把上面的 according to 换成 by Since it follows 2 如果一个论断可以通过一些简单运算或简单推理而获得 由于这些运算或推理比较如果一个论断可以通过一些简单运算或简单推理而获得 由于这些运算或推理比较 简单 读者可以自行推算 因而只需直接写出论断来 这时可用下面句型 简单 读者可以自行推算 因而只需直接写出论断来 这时可用下面句型 1 It is easy to see that It is easy to show that It is easy to prove that It is easy to verify that It is easy to check that 2 It can easily be seen that It can easily be shown that It can easily be proved that It can easily be verified that It can easily be checked that 3 如果所要提及的结论比较显浅 或是众所周知 无需作进一步的证明 这时可用下如果所要提及的结论比较显浅 或是众所周知 无需作进一步的证明 这时可用下 面句型 面句型 1 It is clear that It is obvious that It is evident that It is well known that 2 Clearly Obviously Evidently 4 为了证明一个定理有时需要引进辅助函数 这时可用下面句型 为了证明一个定理有时需要引进辅助函数 这时可用下面句型 Let us first define the function Let us introduce a new function Let us consider the function Let us first investigate the function Let Set Define Put Consider 5 在一个定理中 有几个结论需要证明 其中有些结论比较明显 可不用证明 仅需在一个定理中 有几个结论需要证明 其中有些结论比较明显 可不用证明 仅需 证明余下结论即可 这时可用下面句型 证明余下结论即可 这时可用下面句型 Since A and B are obvious we need only prove C Since A and B are trivial we need only prove C Since A and B are trivial it suffices to prove C 6 为了证明一个定理 有时我们并不是直接去证明 而是证明一个新的论断 一旦新为了证明一个定理 有时我们并不是直接去证明 而是证明一个新的论断 一旦新 的论断得到证明 已给定理不难由此而得证 这时可用下面句型 的论断得到证明 已给定理不难由此而得证 这时可用下面句型 以下各句用于新的论断被证明之前 The theorem will be proved if we can show The result will be proved if we can show The theorem will be proved by showing that If we can prove then the theorem follows immediately 以下各句用于新的论断被证明之后 The theorem is now a direct consequence of what we have proved The theorem follows immediately from what we have proved The theorem is now evident from what we have proved It is evident to see that the theorem holds 7 在证明过程中 有时要用到一些早已学过的知识或技巧 这时可用下面句子 以提在证明过程中 有时要用到一些早已学过的知识或技巧 这时可用下面句子 以提 醒读者 醒读者 Recall that Notice that Note that Observe that In order to prove the theorem we need the knowledge of In order to obtain the following equation we need 8 如果需要证明的定理的假设条件是一般条件 但是 只要定理在特殊条件下成立 如果需要证明的定理的假设条件是一般条件 但是 只要定理在特殊条件下成立 就不难推出定理在一般条件下也成立 这时仅需要在特殊情况下去证明定理就够了 就不难推出定理在一般条件下也成立 这时仅需要在特殊情况下去证明定理就够了 为此可用下面句型 为此可用下面句型 Without loss of generality we may consider Without loss of generality we may assume Without loss of generality we may prove the theorem in the case It suffices to prove the theorem in the case We need only consider the case For simplicity we may take 9 如果待证的论断可用以前用过的相似的方法或步骤进行证明 则可用下面句型 如果待证的论断可用以前用过的相似的方法或步骤进行证明 则可用下面句型 This theorem can be proved in the same way as shown before This statement can be proved in a similar way as shown before This theorem can be proved by the same method as employed in the last section This theorem can be completed by the method analogous to that used above Using the same argument as in the proof of theorem N we can easily carry out the proof of this theorem We now proceed as in the proof of theorem N We shall adopt the same procedure as in the proof of theorem N 10 如果我们用的是反证法 则其开头及结尾可用下面句型 如果我们用的是反证法 则其开头及结尾可用下面句型 If the statement or assertion conclusion were false or not true not right then If the assertion would not hold then This is contrary to This contradicts the fact that This leads to a contradiction 11 表示定理已证毕或者把前面所证的总结为一结论表示定理已证毕或者把前面所证的总结为一结论 We have thus proved the theorem This completes the proof The proof of the theorem is now completed It is now obvious that the theorem holds Thus we have derived that Consequently we infer that Thus we conclude that Thus we are led to the conclusion that Thus we arrive at the conclusion that Thus we can summarize what we have proved as the following theorem 12 其它其它 There exist s such that We claim in fact We are now in a position to If otherwise Provided that 第九章第九章 微分方程微分方程 Chapter12 Differential Equation 解微分方程 solve a dirrerential equation 常微分方程 ordinary differential equation 偏微分方程 partial differential equation PDE 微分方程的阶 order of a differential equation 微分方程的解 solution of a differential equation 微分方程的通解 general solution of a differential equation 初始条件 initial condition 微分方程的特解 particular solution of a differential equation 初值问题 initial value problem 微分方程的积分曲线 integral curve of a differential equation 可分离变量的微分方程 variable separable differential equation 隐式解 implicit solution 隐式通解 inplicit general solution 衰变系数 decay coefficient 衰变 decay 齐次方程 homogeneous equation 一阶线性方程 linear differential equation of first order 非齐次 non homogeneous 齐次线性方程 homogeneous linear equation 非齐次线性方程 non homogeneous linear equation 常数变易法 method of variation of constant 暂态电流 transient stata current 稳态电流 steady state current 伯努利方程 Bernoulli equation 全微分方程 total differential equation 积分因子 integrating factor 高阶微分方程 differential equation of higher order 悬链线 catenary 高阶线性微分方程 linera differential equation of higher order 自由振动的微分方程 differential equation of free vibration 强迫振动的微分方程 differential equation of forced oscillation 串联电路的振荡方程 oscillation equation of series circuit 二阶线性微分方程 second order linera differential equation 线性相关 linearly dependence 线性无关 linearly i