これまで微分の意味と、dy/dxをはじめとする様々な表記方法を見てきました。 これは次の質問につながります。 dxやdyは単体ではどのような意味を持つのでしょうか。 これは前回も触れましたが、まだまだ言いたいことがたくさんあって、そこには収まりきれません。
関数としての微分記号
まず、前回の回答で 2 人が参照した、1998 年に作成されたページから始めます。 この質問は、明確な文脈を持たないため、どのような種類の機能が視野に入っているのか、また、差分に対するどのようなアプローチが取られているのかを示していません。 関数の差分を求める」とはどういうことなのでしょうか？ Jerry 博士は、考えられる文脈の 1 つを提案し、差分が単なる無限小の数であるという、これまでの定義とはまったく異なる定義を与えて、答えました。
Hi Maria,The standard definition of the differential of a real-valued function f of a real variable is: At a given point x, the differential df_x (df sub x; usually the x is omitted) of f is the linear function defined on R by: df_x(h) = f'(x) * hEveryday usage of the differential often suppresses the fact that the differential is a linear function. For example, if y = f(x) = x^2, then we write: dy = df = 2x * dxwhere dx is used instead of h. This is for good reason. The finite numbers dy and dx appearing in dy = 2x * dx can be manipulated to obtain: dy/dx = 2xI feel that I haven't replied directly to your question. I think that this is because I don't fully understand your question. Please write again if my answer has not helped.
この定義では、関数の微分はそれ自体が関数であると考えます。すなわち、与えられた水平方向の変化 (h または ˶ˆ꒳ˆ˵ ) に対する接線に沿った垂直方向の変化 (˶ˆ꒳ˆ˵ ) を値とする関数です。
彼の例では、x = 3における˶‾᷅˵‾᷅˵の微分は、˶‾᷅˵‾᷄˵‾᷅˵‾᷅˵となります。 このように考えると、微分を数値のように書くという通常の方法は、単なる近道でしかない。 変数 x を残したまま、完全には ˶ˆ꒳ˆ˵ ( df_x(dx) = 2x dx˶ ) 、簡潔には ˶ˆ꒳ˆ˵ ( dy = 2x dx˶ ) となります。
Maria は、もう少し文脈を与えながらも、自分がどのレベルにいるのかを明確にしないまま、さらに質問してきました。
Thanks for your answer. I know that the question is a little bit confusing, and at the beginning I thought it was a problem of the translation from English of the Math books. Your answer helped a little, so I am going to try to rephrase it.What is the difference between finding the derivatives of a function (dy/dx), and finding its differentials (dy, dx)?In the books I've seen they define differentials supposing that f(x) is differentiable.My teacher gave a hint to reach this conclusion: if you can find the differentials of f, then f is differentiable, but if f is differentiable you can't necessarily find its differentials.That is why I can prove this, starting with a function that is differentiable.
「関数の導関数」が何を意味するのか、まだ不明です。
ドクター・ジェリーは、前回の定義を再確認することで回答を始めました：
Hi Maria,Suppose f(x) = x^2. To find the derivative of f we use the definition of derivative: f'(x) is the limit as h->0 of the quotient f(x+h) - f(x) ------------- hFor this function, f'(x) = 2x.Okay, this much is clear; there is no possible ambiguity.The differential of f at x is defined to be the linear function df, which is defined on all of R by: df(h) = f'(x) * hOften, the notation df(h) is shortened to df or, if y = f(x), then we write dy instead of df. Then the above definition is: dy = f'(x)*dx or dy/dx = f'(x)Unless you are studying differential geometry, in which dx is interpreted slightly differently, dx is not the differential of a function. It is a variable, the same as h.
質問とその文脈が明確にされていないと思うので、どのような回答が必要なのか明確ではないので、残りの回答は省略します。
もっと深く掘り下げたい場合…
ドクター・ジェリーは、微分がより深く定義される場所として、微分幾何学に一瞥して言及しました。
There is also a more sophisticated viewpoint in which what is integrated is not a function f(x), but rather what is called a "differential form". This viewpoint involves a lot of complicated mathematical structure and is more commonly seen in calculus of functions of several variables (see, for example, http://en.wikipedia.org/wiki/Differential_form )but it can also be used in one-dimensional calculus as well (e.g. in David Bressoud's book _Second Year Calculus_).So, the easiest viewpoint is the purely formal one, in which you do useful but basically meaningless computations (du=g'(x)dx which does the bookkeeping), but there is also a more complicated viewpoint in which the computations are not meaningless, but they require you to learn more abstract mathematics. For example, the one-dimensional differential form dx becomes a mapping from intervals on the real line to R, and dx() = b-a ,while the differential form 3x^2dx (to use one of Bressoud's examples) is the mapping which takes the interval to b / b^3 a^3 | 3x^2 dx = --- - --- . / 3 3 aThis becomes the viewpoint used in modern differential geometry.
定積分記法における微分
先週は微分の記号の中での微分の使用について話しました。 今回は、積分での使用に関するいくつかの質問を見てみましょう。
The Meaning of 'dx' in an IntegralNo matter how many times it's explained to me, and even though I've taken several advanced math courses (diff eq, linear algebra, etc), nobody has ever given me a satisfactory explanation for the meaning of the notation in which an integral has dx appended to the end if x is the variable which we are integrating with respect to. In physics, for example, dx seems to mean a very small amount of x, and then we use it in an integral to integrate whatever physical quantity is being discussed. I just don't understand. Or, when a differential is defined, all of a sudden the dx has a meaning, but then when an integral is being evaluated, the teacher says, "Oh, the dx is just a formality." So, sometimes it's a formality, sometimes a vital concept, sometimes a physical quantity, sometimes a derivative: What is it?
「\int f(x) dx\」と書くとき、私たちはこれを「f(x)のxに対する積分」と読んでいますが、「dx」にはどの変数を気にするかということ以外の意味はありません。 (実際には、変数がはっきりしている場合には、dxを完全に省略できることもあります!) これは、「xに関して」という意味で使われる微分の場合とあまり変わりません。 ここではどのような意味があるのでしょうか。
Dr. Jeremiahは、定積分のアイデアに焦点を当てて質問を受けました：
Hi Nosson,Think about it this way:An integral gives you the area between the horizontal axis and the curve. Most of the time this is the x axis. y | | --|-- ----|---- f(x) / | \ / | / | -------- | | / | | -----|------- | | | | | | | | ----------|--------------+--------------------|----- x a bAnd the area enclosed is: b / Area = | f(x) dx / a
これは広い意味での定積分の定義です。
But say you didn't want to use an integral to measure the area between the x axis and the curve. Instead you just calculate the average value of the graph between a and b and draw a straight flat line y = avg(x) (the average value of x in that range).Now you have a graph like this: y | | - | - - - | - - f(x) | / | \ / | -----|-----------------------------------|---- avg(x) | / | | - - -|- - - - | | | | | | | | ----------|--------------+--------------------|----- x a bAnd the area enclosed is a rectangle: Area = avg(x) w where w is the width of the sectionThe height is avg(x) and the width is w = b-a or in English, "the width of a slice of the x axis going from a to b."
彼の幅wはしばしば\(Delta x\)と呼ばれますが、これは後述します。
But say you need a more accurate area. You could break the graph up into smaller sections and make rectangles out of them. Say you make 4 equal sections: y | | |----|---| |-------|---- f(x) | | | | | | | |--------| | | | | | | | -----|---------| | | | | | | | | | | | | | | | | ----------|---------|----+---|--------|-------|----- x a bAnd the area is: Area = section 1 + section 2 + section 3 + section 4 = avg(x,1) w + avg(x,2) w + avg(x,3) w + avg(x,4) wwhere w is the width of each section. The sections are all the same size, so in this case w=(b-a)/4 or in English, "the width of a thin slice of the x axis or 1/4 of the width from a to b."
ライマン積分を開発し始めています (ただし、幅が実際には同じである必要はないなど、完全な定義を行うには多くの詳細が必要です)。
And if we write this with a summation we get: 4 +--- \ Area = / avg(x,n) w +--- n=1But it's still not accurate enough. Let's use an infinite number of sections. Now our area becomes a summation of an infinite number of sections. Since it's an infinite sum, we will use the integral sign instead of the summation sign: / Area = | avg(x) w /where avg(x) for an infinitely thin section will be equal to f(x) in that section, and w will be "the width of an infinitely thin section of the x axis."So instead of avg(x) we can write f(x), because they are the same if the average is taken over an infinitely small width.
ここでも、直感的に理解できるように多くの詳細を省略しています。
And we can rename the w variable to anything we want. The width of a section is the difference between the right side and the left side. The difference between two points is often called the delta of those values. So the difference of two x values (like a and b) would be called delta-x. But that is too long to use in an equation, so when we have an infinitely small delta, it is shortened to dx.If we replace avg(x) and w with these equivalent things: / Area = | f(x) dx /
つまり、微分に対する無限小のアプローチのように、dxは（非公式に）xの非常に小さな変化と考えられます。
So what the equation says is:Area equals the sum of an infinite number of rectangles that are f(x) high and dx wide (where dx is an infinitely small distance).So you need the dx because otherwise you aren't summing up rectangles and your answer wouldn't be total area.dx literally means "an infinitely small width of x".
これはもちろん、定積分に特に当てはまります。 この観点から、不定積分は微積分の基本定理によって同じ表記法を継承していると考えることができ、この2つを結び付けています。
Integral Notation - Missing IntegrandsI have seen some integral notation used that I am not familiar with. It looks like this: / | dx f(x) + .../There does not seem to be an integrand (i.e. a function being integrated). I'm not sure if f(x) is to be integrated. I have two theories, but I can't see the point in writing the expression as it is if either of my theories is correct.My theories about what this might mean:1) The above notation is the same as writing: / | 1 dx f(x) + ... (note the explicit 1 here)/=(x + C) * f(x) + ... (where C is a constant of integration)2) The rest of the expression is to be integrated with respect to x.If (1) is correct, then what was the point of writing the integral - why wasn't (x + C) just written instead? If (2) is correct, then how does one know when to "stop integrating" (i.e. if there is some term to be added on to the expression that is not to be integrated, how is it distinguished?).I have seen this recently in multi-variate calculus, i.e. when x is in R^n rather than R: does this situation justify the use of the integral notation somehow?
Chris の最初の推測は、dx が積分を閉じているので、後に続くものは乗算されなければならないというものでした。
この表記法が、複数の変数を持つ微積分で特に一般的であることは、彼の言うとおりです。 例えば、$int_0^b dy\int_0^a dx f(x,y)$、$int_0^b dy\int_0^a f(x,y)dx dx$ではなく、$$int_0^b dy\int_0^a f(x,y)dx dy$と書くことで、まずxに関して積分し、その結果をyに関して積分することを示すことができます。
私の回答：
Hi, Chris.It is common to learn about integration in such a way that the "dx" seems to be a marker for the end of the integral, as if the "long S" were a left parenthesis and the "dx" were the right parenthesis. But it doesn't work that way. In fact, what you are integrating is the product of a function and dx; and multiplication is commutative! So these mean the same thing: / / | f(x) dx and | dx f(x) / /If you then add something, you must use parentheses if it is to be part of the integral: / / | dx f(x) + g(x) = + g(x) / /is the sum of an integral and a function, while / / | dx (f(x) + g(x)) = | (f(x) + g(x)) dx / /is the integral of the sum of two functions.That is, presumably the integral has higher precedence than addition, so you "stop integrating" at the first plus sign. But even then, I'm not positive that this rule I just made up is always followed; let me know if you think it doesn't fit the practice in your text, and show me an example.
微分を積の一部として見ることは、表記法を理解するために必要です。
括弧についての私の考えが普遍的に守られているわけではありませんが、 \(int (x^2-2x+3) dx\)ではなく、 \(int x^2-2x+3 dx\)と表示されることは珍しくありません。 これは、微分を積分の最後に使うのが一般的であることと、通常の操作順序にもかかわらず、dxを最後の項にのみ関連するものとみなすのは無意味であるという事実によるものです。 このようないい加減さは、dxが最初に書かれる積分にも引き継がれるかもしれませんが、そこでは曖昧さがより大きくなります。
これを書いていて、特に定積分に関しては、可換性への私の言及はあまり有効ではないことに気付きました。 $$int_0^b\int_0^a f(x,y)dx dy\int_0^a f(x,y)dy dx$
それは、微分の順序が積分の限界の意味を決めるからです。
Chrisが答えてくれました
Doctor Peterson,Thank you for your quick and helpful reply.I was indeed taught that integration begins with the "long S" and ends with the (for example) dx.I have, however, seen the following notation: / | dx | ------------ | f(x) + g(x)/and assumed it was a convenient notation rather than being a justifiable mathematical expression.Perhaps I need to go and look at calculus from first principles again to see why this is the case.
それは便利な記法であり、また正当なものです!
不定積分における微分の有用性を示す特に良い例が置換法で、dxを実際に乗算する式で置き換えることができます。