Mathematics Yoshio Sep 8th, 2023 at 8:00 PM 8 0

## 微積分 Calculus

Calculus#### Fundamental Theorem of Calculus

##### Integration

\(g(t)=\int_a^tf(\tau)\operatorname d\tau,\;(a\leq t\leq b)\)

\(\dot g(t)=f(t)\)

\(\dot g(t)=\lim_{\triangle t\rightarrow0}\frac{g(t+\triangle t)-g(t)}{\triangle t}\)

\(=\lim_{\triangle t\rightarrow0}\frac{f(t)\triangle t}{\triangle t}=f(t)\)

##### Antiderivative

\(\int_a^bf(\tau)\operatorname d\tau=h(b)-h(a)\)

\(h(t)=\int_\alpha^tf(\tau)\operatorname d\tau\)

\(\dot h(t)=f(t)\)

\(h(b)-h(a)=\int_\alpha^bf(\tau)\operatorname d\tau-\int_\alpha^af(\tau)\operatorname d\tau=\int_a^bf(\tau)\operatorname d\tau\)

Ex

\(g(t)=\int_a^tf(\tau,t)\operatorname d\tau\)

\(\dot g(t)=f(t,t)+\int_a^t\frac\partial{\partial t}\int f(\tau,t)\operatorname d\tau\)

\(f(\tau,t)=t^2+2\tau t\)

\(g(t)=\int_{-3}^t(t^2+2\tau t)\operatorname d\tau\)

\(g(t)=\left.(t^2\tau+t\tau^2)\right|_{-3}^t\)

\(=2t^3+3t^2-9t\)

\(\dot g(t)=6t^2+6t-9\)

(a) \(f(t,t)=t^2+2t^2=3t^2\)

\(\frac\partial{\partial t}f(\tau,t)=2t+2\tau\)

(b) \(\int_{-3}^t\frac\partial{\partial t}f(\tau,t)\operatorname d\tau=\int_{-3}^t(2t+2\tau)\operatorname d\tau\)

\(=\left.(2t\tau+\tau^2)\right|_{-3}^t=2t^2+t^2+6t-9\)

(a)+(b) \(\dot g(t)=6t^2+6t-9\)

### Derivatives Functions Rules

##### The Sum and Difference Rules

\( \frac d{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac d{dx}f\left(x\right)+\frac d{dx}g\left(x\right)\)

\( \frac d{dx}\left[f\left(x\right)-g\left(x\right)\right]=\frac d{dx}f\left(x\right)-\frac d{dx}g\left(x\right)\)

\(h\left(x\right)=f\left(x\right)+g\left(x\right)\)

\(\frac d{dx}h\left(x\right)\)

\(=\lim\limits_{\triangle x\rightarrow\infty}\frac{h\left(x+\triangle x\right)-h\left(x\right)}{\triangle x}\)

\(=\lim\limits_{\triangle x\rightarrow\infty}\frac{\left[f\left(x+\triangle x\right)+g\left(x+\triangle x\right)\right]-\left[f\left(x\right)+g\left(x\right)\right]}{\triangle x}\)

\(=\lim\limits_{\triangle x\rightarrow\infty}\frac{\left[f\left(x+\triangle x\right)-f\left(x\right)\right]+\left[g\left(x+\triangle x\right)-g\left(x\right)\right]}{\triangle x}\)

\(=\lim\limits_{\triangle x\rightarrow\infty}\frac{f\left(x+\triangle x\right)-f\left(x\right)}{\triangle x}+\lim\limits_{\triangle x\rightarrow\infty}\frac{g\left(x+\triangle x\right)-g\left(x\right)}{\triangle x}\)

\(=\frac d{dx}f\left(x\right)+\frac d{dx}g\left(x\right)\)

##### The Product Rule

\(\frac d{dx}\left[f\left(x\right)\cdot g\left(x\right)\right]=g\left(x\right)\cdot\frac d{dx}f\left(x\right)+f\left(x\right)\cdot\frac d{dx}g\left(x\right)\)

##### The Quotient Rule

\(D{\textstyle\left({\displaystyle\textstyle\frac fg}\right)}=\frac{g\cdot D\left(f\right)-f\cdot D\left(g\right)}{g^2}\)

\(D{\textstyle\left({\displaystyle\textstyle\frac fg}\right)}\)

\(=D\left(f\cdot g^{-1}\right)\)

\(=f\cdot D\left(g^{-1}\right)+D\left(f\right)\cdot g^{-1}\)

\(=-f\cdot g^{-2}\cdot D\left(g\right)+D\left(f\right)\cdot g^{-1}\)

\(=-f\cdot g^{-2}\cdot D\left(g\right)+D\left(f\right)\cdot g\cdot g^{-2}\)

\(=\frac{g\cdot D\left(f\right)-f\cdot D\left(g\right)}{g^2} \)

##### The Chain Rule

\[\frac{dy}{dx}=\frac{dy}{du}{}\frac{du}{dx}\]

Suppose that we have two functions \(f(x)\) and \(g(x)\) and they are both differentiable.

If we define \(F(x) = (f∘g)(x)\) then the derivative of \(F(x)\) is,

\[F^{\prime}{(x)}=f^{\prime}{({g{(x)}})}{}g^{\prime}{(x)}\]

If we have \(y = f(u)\) and \(u = g(x)\) then the derivative of \(y\) is,

\[\frac{dy}{dx}=\frac{dy}{du}{}\frac{du}{dx}\]

## Partial Integration

\(\frac d{dx}\lbrack f(x)\cdot g(x)\rbrack\;=\;g(x)\cdot\frac d{dx}f(x)+f(x)\cdot\frac d{dx}g(x)\)

\(\int\left\{\frac d{dx}\lbrack f(x)\cdot g(x)\rbrack\right\}\;dx\)

\(=\int\lbrack g(x)\cdot\frac d{dx}f(x)+f(x)\cdot\frac d{dx}g(x)\rbrack\;dx\)

\(=\int\lbrack g(x)\cdot\frac d{dx}f(x)\rbrack dx+\int\lbrack f(x)\cdot\frac d{dx}g(x)\rbrack dx\)

\(f(x)\cdot g(x)=\int\lbrack g(x)dx\rbrack\frac d{dx}f(x)+\int\lbrack f(x)dx\rbrack\frac d{dx}g(x)\)

\(f\cdot g\;=\;\int g\cdot\operatorname df+\int f\cdot\operatorname dg\)

\(\int f\cdot\operatorname dg\;=f\cdot g-\;\int g\cdot\operatorname df\)

\(\int u\operatorname dv=uv-\int v\operatorname du\)

### Tabular Integration

Repeated integration by parts, also called the DI method of Hindu method

Considering a second derivative of \(v\) in the integral on the LHS of the formula for partial integration suggests a repeated application to the integral on the RHS:

Extending this concept of repeated partial integration to derivatives of degree \(n\) leads to

Step | Procedure |
---|---|

Step 1 | In the product comprising the function \(f\), identify the polynomial and denote it \(F(x)\). Denote the other function in the product by \(G(x)\). |

Step 2 | Create a table of \(F(x)\) and \(G(x)\), and successively differentiate F(x) until you reach \(0\). Successively integrate \(G(x)\) the same amount of times. |

Step 3 | Negate every second entry under \(F(x)\). |

Step 4 | Construct the integral by taking the product of \(F(x)\) and the first integral of \(G(x)\), then add the product of \(F′(x)\) times the second integral of \(G(x)\), then add the product of \(F′′(x)\) times the third integral of \(G(x)\), etc… |

##### LIATE rule

Whichever function comes first in the following list should be \(u\):

Abbreviation | FunctionType | Examples |
---|---|---|

L | Logatithmic functions | \(\ln(x),\;\log_2(x),\;etc\) |

I | Inverse trigonometric functions | \(\tan^{-1}(x),\;\sin^{-1}(x),\;etc\) |

A | Algebraic functions | \(x,\;3x^3,\;5x^{21},\;etc\) |

T | Trigonometric functions | \(\cos(x),\;\tan(x),\;sech(x),\;etc\) |

E | Exponential functions | \(e^x,\;7^x,\;etc\) |

##### Partial Integration Applications

Antiderivatives

Polynomials and trigonometric functions

Exponentials and trigonometric functions

Gamma function identity

### Green's first identity

https://en.wikipedia.org/wiki/Integration_by_parts

Integration by parts can be extended to functions of several variables by applying a version of the fundamental theorem of calculus to an appropriate product rule. There are several such pairings possible in multivariate calculus, involving a scalar-valued function u and vector-valued function (vector field) V.