phrases or clauses frequently used in mathematics
常見的涉及運算的短語或句子 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)為某些表達式，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 form12. by a simple calculation, we obtain from (a) … 定理證明過程中常見的短語和句子 1.下面的句型可用來表達“根據(jù)什么即可得到什么”的意思according to definition , it follows … according to hypothesis , it follows … according to asssumptions, 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 ”換成“ by” )since …, it follows … 2. 如果一個論斷可以通過一些簡單運算或簡單推理而獲得，由于這些運算或推理比較簡單，讀者可以自行推算，因而只需直接寫出論斷來，這時可用下面句型：(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.如果所要提及的結(jié)論比較顯淺，或是眾所周知，無需作進一步的證明，這時可用下面句型： (1) it is clear that …it is obvious that …it is evident that …it is well-known that …(2) clearly, … obviously, … evidently,… 4.為了證明一個定理有時需要引進輔助函數(shù)，這時可用下面句型： 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. 在一個定理中，有幾個結(jié)論需要證明，其中有些結(jié)論比較明顯，可不用證明，僅需證明余下結(jié)論即可，這時可用下面句型： 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. 為了證明一個定理，有時我們并不是直接去證明，而是證明一個新的論斷，一旦新的論斷得到證明，已給定理不難由此而得證，這時可用下面句型： 以下各句用于新的論斷被證明之前：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.以下各句用于新的論斷被證明之后：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.在證明過程中，有時要用到一些早已學過的知識或技巧，這時可用下面句子，以提醒讀者： 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. 如果需要證明的定理的假設條件是一般條件，但是，只要定理在特殊條件下成立，就不難推出定理在一般條件下也成立，這時僅需要在特殊情況下去證明定理就夠了，為此可用下面句型： 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. 如果待證的論斷可用以前用過的相似的方法或步驟進行證明，則可用下面句型： 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. 如果我們用的是反證法，則其開頭及結(jié)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. 表示定理已證畢或者把前面所證的總結(jié)為一結(jié)論 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 there exist(s)…such that… we claim…in fact… we are now in a position to… if otherwise… provided that…