One thing I never learned was the integration of contours. I had learned to make integrals using various methods, which were shown in a book given to me by my physics teacher, Bader. One day he told me to stay after class. «Feynman,» he said, «you talk too much and make too much noise. I know why. You`re bored. So I`ll give you a book. You go to the back of the corner, and study this book, and when you know everything in this book, you can speak again. So in every physics class, I didn`t pay attention to what was going on with Pascal`s law, or whatever they did. I was late with this book: Woods` Advanced Calculus. Bader knew that I had studied a bit of «calculus for the practical man», so he gave me the real job – it was for a junior or senior course at university. There were Fourier series, Bessel functions, determinants, elliptic functions – all sorts of wonderful things that I didn`t know about. This book also showed how to distinguish parameters under the integral sign — this is a specific operation.
It turns out that this is not taught much in universities; They don`t stress him out. But I figured out how to use this method, and I used this damn tool over and over again. So, because I was self-taught with this book, I had strange methods for making integrals. The result was that if the boys at MIT or Princeton struggled to do some integral, it was because they couldn`t do it using the standard methods they had learned in school. If it were an integration of contours, they would have found it; If it was a simple series expansion, they would have found it. Then I come and try to distinguish under the integral sign, and often it has worked. So I had a good reputation for integrals simply because my toolbox was different from everyone else`s, and they had tried all their tools before giving me the problem. This is the general form of Leibniz`s integral rule. The rule for Leibniz`s formula for the product of two functions f(x), g(x) is (dfrac{d}{dx}.f(x).g(x) = g(x).dfrac{d}{dx}.f(x) + f(x).dfrac{d}{dx}.g(x)). This rule can be used to find some unusual specific integrals such as The generalized formula of Leibniz`s rule ((f(x).g(x))^n = sum^nC_rf^{(n — r)}(x).g^r(x)) can be simplified to find the first derivative of the product of two functions like (f(x).g(x))` = f`(x).g(x) + f(x).g`(x), and the second derivative of the product of two functions like (f(x).g(x)« = f«(x).g(x) + 2f`(x).g`(x) + f(x).g`(x). Leibniz`s rule can be used for the product of two or more functions and for n derived from the product of functions.
If a ( x ) = a {displaystyle a(x)=a} is constant and b ( x ) = x {displaystyle b(x)=x} , which is another common situation (for example, in the proof of Cauchy`s repeated integration formula), Leibniz`s integral rule: Leibniz`s rule can be applied to the product of several functions and for many derivatives. Let`s understand the different formulas and proofs of Leibniz`s rule. Therefore, the derivative of the product of two functions using Leibniz`s rule ( 2Tanx + 4x. Sek^2x+2x^2Sek^2x. Tanx). The general statement of Leibniz`s integral rule requires concepts of differential geometry, especially differential forms, external derivatives, corner products, and interior products. With these tools, Leibniz`s integral rule in n dimensions [4] Therefore, the derivative of the product of the two given functions (4x^3.Logx + x^3) A Leibniz integral rule for a two-dimensional surface moving in three-dimensional space is[3][4] The general form of Leibniz`s integral rule with variable limits can be derived as a consequence of the basic form of Leibniz`s integral rule, The multivariate chain rule and the first fundamental theorem of calculus. For example, suppose f {displaystyle f} is defined in a rectangle in the plane x − t {displaystyle x-t}, for x ∈ [ x 1 , x 2 ] {displaystyle xin [x_{1},x_{2}]} and t ∈ [ t 1 , t 2 ] {displaystyle tin [t_{1},t_{2}]}. For example, suppose f {displaystyle f} and the partial derivative ∂ f ∂ x {textstyle {frac {partial f}{partial x}}} are both continuous functions on this rectangle. Suppose a , b {displaystyle a,b} are differentiable real-valued functions defined on [ x 1 , x 2 ] {displaystyle [x_{1},x_{2}]} , with values in [ t 1 , t 2 ] {displaystyle [t_{1},t_{2}]} (i.e. for each x ∈ [ x 1 , x 2 ] , a ( x ) , b ( x ) ∈ [ t 1 , t 2 ] {displaystyle xin [x_{1}, x_{2}],a(x),b(x)in [t_{1},t_{2}]} ). This now follows from the chain rule and the first fundamental theorem of calculus.
Definition Leibniz`s rule for the product of two functions is (f(x).g(x))« = f«(x).g(x) + 2f`(x).g`(x) + f(x).g`(x). To determine C 2 {displaystyle C_{2}} in the same way, we need to replace a value of α {displaystyle alpha } greater than 1 in φ ( α ) {displaystyle varphi (alpha )}. It`s a bit heavy. Instead, we replace α = 1 β {textstyle alpha ={frac {1}{beta }}} where | β | < 1 {displaystyle |beta |<1}. This important result can be used under certain conditions to exchange operators to integral and partial derivatives, and is particularly useful for differentiating integral transformations. An example of this is the moment-generating function in probability theory, a variation of the Laplace transform that can be differentiated to generate the moments of a random variable. Whether Leibniz`s integral rule applies is essentially a question of border exchange. Stronger versions of the theorem only require that the partial derivative exist almost everywhere, not that it be continuous. [2] This formula is the general form of Leibniz`s integral rule and can be derived using the fundamental theorem of calculus.
The (first) fundamental theorem of infinitesimal calculus is only the special case of the above formula, where a ( x ) = a ∈ R {displaystyle a(x)=ain mathbb {R} } , b ( x ) = x {displaystyle b(x)=x} , and f ( x , t ) = f ( t ) {displaystyle f(x,t)=f(t)}. Leibniz`s rule can be proved using mathematical induction. Let f(x) and g(x) n times be the distinguishable functions. If we apply the initial case of mathematical induction for n = 1, we obtain the following expression. This is the simple product rule that applies to n = 1. Suppose this statement is true for all n > 1, and we have the following expression. Leibniz`s integral rule gives a formula for differentiating a particular integral whose limits are functions of the differential variable, Leibnitz`s theorem is basically Leibnitz`s rule defined for the derivative of the antiderivative. According to the rule, the nth order derivative of the product of two functions can be expressed using a formula. Functions that could probably have given a function as a derivative are called antiderivatives (or primitives) of the function. The formula that gives all these antiderivatives is called the indefinite integral of the function, and such a process of searching for antiderivatives is called integration.
Now let`s discuss the formula and proof of Leibnitz`s rule here. Leibniz`s rule for the first derivative of the product of the functions f(x) and g(x) is (f(x).g(x))` = f`(x).g(x) + f(x).g`(x) or (dfrac{d}{dx}.f(x).g(x) = g(x).dfrac{d}{dx}.f(x) + f(x).dfrac{d}{dx}.g(x)), or (dfrac{d}{dx}.f(x).g(x) = g(x).dfrac{d}{dx}.f(x) + f(x).dfrac{d}{dx}.g(x)). The sign of the line integral is based on the legal rule for choosing the direction of the line element ds. For example, to establish this sign, suppose that the field F points in the positive z direction and that the surface Σ is a part of the xy plane of circumference ∂Σ. We assume that the normal to Σ is in the positive z-direction. The positive traversal of ∂Σ then occurs counterclockwise (ruler on the right with the thumb along the z-axis). Then the left integral determines a positive flow of F by Σ. Suppose that Σ translates in the positive x direction at velocity v. An element of the boundary of Σ parallel to the y-axis, for example ds, sweeps a surface vt × ds at time t. If we integrate around the boundary ∂Σ counterclockwise, vt × ds points in the negative z direction on the left side of ∂Σ (where ds points down) and in the positive z direction on the right side of ∂Σ (where ds points upwards), which makes sense because Σ moves to the right, Adds surface area to the right and loses to the left. On this basis, the flux of F increases to the right of ∂Σ and decreases to the left. The point product v × F ⋅ ds = −F × v ⋅ ds = −F ⋅ v × ds.
Therefore, the sign of the straight line integral is assumed to be negative. Leibniz`s rule is used to find the first, second, or derivative of the product from two or more functions. Leibniz`s rule for the first derivative of the product of two functions is (f(x).g(x))` = f`(x).g(x) + f(x).g`(x), and Leibniz`s rule for the second derivative of the product of two functions is (f(x).g(x)« = f«(x).g(x) + 2f`(x).g`(x) + f(x).g`(x).
