Partial Differentiation and Multiple Integral

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Partial Differentiation and Multiple Integral

發表  dalcde 于 周二 7月 06, 2010 8:06 pm

After normal differentiation, we can use integration to retrieve back the original function? After partial differentiation, is it possible to retrieve the original function back? If it is possible, how?

At wikipedia, it is said a multiple integral is "a type of definite integral extended to functions of more than one real variable". Is there another integral which is similar to the multiple integral for indefinite integral? (which I think is the same thing as what I was asking at the first question)
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發表  clovis_szeto 于 周二 7月 06, 2010 11:10 pm

HAHA, you better don't talk about it in M1, otherwise my students will feel confused on what they have learnt....HAHA

For the 1st question, After normal differentiation, we can use integration to retrieve back the original function?
--> the answer is no
as you know if the dy/dx=f(x) then y= integrate f(x) +C <-- there will be a constant behind, which may translate the curve upwards and downwards, but it will not be the same function.

2nd question:After partial differentiation, is it possible to retrieve the original function back? If it is possible, how?
Answer:
As some people don't know what's meant by partial differentiation, so if you would like to know, you can take a look at http://en.wikipedia.org/wiki/Partial_derivative,
The following is just my opinion, i haven't proved it
for a function z=f(x,y), if you get both the derivative with respect to x and y, then you can find a similar figure (3D), but still you can't find the exact figure, the same reason as q1, there are 2 constants created by in x and in y.

question3: HAHA, i haven't heard about it ......

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發表  dalcde 于 周四 7月 08, 2010 9:03 pm

Thanks. By the way, what is
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發表  clovis_szeto 于 周四 7月 08, 2010 10:24 pm

you can't integrate it, can't find the answer!!!

Actually you can differentiate nearly all the function, but the reverse case is totally different!!

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