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Advanced notes (Part 2)

2022-07-28 20:01:00 【Ordinary konjaku 99】

The first 8 Chapter Space analytic geometry and vector algebra

1、 The formula for the distance between two points ：

$d=\sqrt{(x2-x1)^{2}+(y2-y1)^{2}+(z2-z1)^{2}}$

2、 Bearing 、 Directional cosine ：

vector $\vec{r}=(x,y,z)$ ,α,β,γ It is called the direction angle , Then the direction cosine $cos\alpha =\frac{x}{|\vec{r}|}$ , $cos\beta =\frac{y}{|\vec{r}|}$ , $cos\gamma =\frac{z}{|\vec{r}|}$

$cos ^{2}\alpha+cos ^{2}\beta +cos^{2}\gamma =1$ , It is used to judge whether a group of angles is the direction angle of a vector

3、 Projection

vector $\vec{r}$ stay $\vec{u}$ The projection on is recorded as Prj u r, $Prj u r=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|}$

It can be used to express the product of quantity $\vec{a}\cdot \vec{b}=|\vec{a}|Prj a b$ （ When $\vec{a}\neq 0$ when ）, $\vec{a}\cdot \vec{b}=|\vec{b}|Prj b a$ （ When $\vec{b}\neq 0$ when ）

4、 The product of quantity 、 Vector product

$\vec{a}=(a1,a2,a3),\vec{b}=(b1,b2,b3)$ , be

The product of quantity $\vec{a}\cdot \vec{b}=|\vec{a}||\vec{b}|cos<\vec{a},\vec{b}>=a1b1+a2b2+a3b3$

Vector product $\vec{a}\times \vec{b}=\begin{vmatrix} i &j &k \\ a1& a2 &a3 \\ b1 &b2 &b3 \end{vmatrix}$

*5、 The mixed product of vectors

$[\vec{a}\vec{b}\vec{c}]=(\vec{a}\times \vec{b})\cdot \vec{c}$ , Its absolute value is expressed in $\vec{a$ 、 $\vec{b}$ 、 $\vec{c}$ The volume of a parallelepiped with edges

6、 Two vectors in space are perpendicular 、 Parallel judgment

(1)、 $\vec{a}\perp \vec{b}\Leftrightarrow \vec{a}\cdot \vec{b}=0$

(2)、 $\vec{a}\parallel \vec{b}\Leftrightarrow \vec{a}\times \vec{b}=0$ namely $\vec{a}=\lambda \vec{b}$

*7、 Formula of fixed ratio and dividing point

Both ends of the line segment A(x1,y1,z1),B(x2,y2,z2),M For its upper point , $\lambda =\frac{|AM|}{|BM|}$ ,

be M Coordinate for $\frac{1}{1+\lambda }(x1+\lambda x2,y1+\lambda y2,z1+\lambda z2)$

8、 Plane and its equation

（1）、 Point to French

A(x-x0)+B(y-y0)+c(z-z0)=0 A little on the face （x0,y0,z0）, The normal vector （A,B,C）

（2）、 General style

Ax+By+Cz+D=0 ( $A^{2}+B^{2}+C^{2}\neq 0$ )

（3）、 Intercept type

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ , a,b,c The planes are respectively x,y,z Axis intercept

9、 Point to face distance formula

spot M（x0,y0,z0）, Noodles Ax+By+Cz+D=0

$d=\frac{|Ax0+By0+Cz0+D|}{\sqrt{A^{2}+B^{2}+C^{2}}}$

10、 Space straight line and its equation

（1）、 General style

$\left\{\begin{matrix} A1x+B1y+C1z+D1=0\\ A2x+B2y+C2z+D2=0 \end{matrix}\right.$

（2）、 Symmetrical （ Pointwise ）

$\frac{x-x0}{m}=\frac{y-y0}{n}=\frac{z-z0}{p}$ , A little online M(x0,y0,z0), The direction of the vector $\vec{s}$ =（m,n,p）

（3）、 Parametric equation

$\left\{\begin{matrix} x=x0+mt\\y=y0+nt \\z=z0+pt \end{matrix}\right.$ , A little online M(x0,y0,z0), The direction of the vector $\vec{s}$ =（m,n,p）

（4）、 Two point equation

$\frac{x-x1}{x2-x1}=\frac{y-y1}{y2-y1}=\frac{z-z1}{z2-z1}$

11、 Several common quadratic surface equations

sphere ： $Ax^{2}+Ay^{2}+Az^{2}+Dx+Ey+Fz+G=0$ (A≠0, $D^{2}+E^{2}+F^{2}>4AG$ )

Ellipsoid ： $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$

Conical surface ： $z^{2}=a^{2}(x^{2}+y^{2})$

Cylindrical surface ： $x^{2}+y^{2}=a^{2}$

Elliptical cylinder ： $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Parabolic cylinder ： $y^{2}=2px$

Conical surface ： $z^{2}=a^{2}(x^{2}+y^{2})$

Rotating hyperboloid ： $\frac{x^{2}}{a^{2}}-\frac{y^{2}+z^{2}}{c^{2}}=1$ Bilobate , $\frac{x^{2}+y^{2}}{a^{2}}-\frac{z^{2}}{c^{2}}=1$ simple leaf

12、 Solution of rotating surface

Who will not change , Other variables become the sum of squares under positive and negative roots

The first 9 Chapter Differential calculus of multivariate functions

1、 Multivariate functions are differentiable 、 Partial derivatives exist 、 The relationship between partial derivative continuity and function continuity ：

2、 Partial derivatives of multivariate composite functions ：

Line multiplication 、 Add lines . Such as :

3、 Total differential

4、 Implicit function for partial derivation

F(x,y,z) = 0,

5、 Tangent and normal plane of space curve

The space curve , stay t=t0 The tangent equation at is , The normal plane equation is , among Is the direction vector of the tangent and the normal vector of the plane

6、 The tangent plane and normal of the surface

（1）、 Surfaces , stay （x0,y0,z0） Situated

The tangent plane equation is ,

The normal equation is , The normal vector of the tangent plane and the direction vector of the normal are

;

（2）、 The surface is z=f(x,y)

Surface in M(x0,y0,z0) The normal vector at is （fx(x0,y0) , fy(x0,y0) , -1）

The tangent plane equation is fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)-(z-z0)=0

The normal equation is $\frac{x-x0}{fx(x0,y0)}=\frac{y-y0}{fy(x0,y0)}=\frac{z-z0}{-1 }$

7、f(x,y) stay （x0,y0） Conditions for obtaining extreme values at

fxx(x0,y0)=A,fxy(x0,y0)=B,fyy(x0,y0)=C

（1） $AC-B^{2}>0$ when , There are extreme values . When A<0 when , There are maxima ; When A>0 when , There is a minimum

（2） $AC-B^{2}< 0$ when , There is no extreme value

（3） $AC-B^{2}= 0$ when , There may be extreme values , There may also be no extreme value , It needs to be discussed separately

8、 Use Lagrange multiplier method to solve simple extreme value application problem ：（ Conditional extremum ）

① Use known conditions and related knowledge to establish functional relationships （ Take the binary function for example ）

And determine the constraints

② Construct Lagrange function

③ from Find the standing point , Combined with the practical significance of the problem itself , Confirm to get The maximum value point and the maximum value of ．

9、 Directional derivative

$\frac{\partial f}{\partial l}|_{(x0,y0)}=fx(x0,y0)cos\alpha +fy(x0,y0)cos\beta$

1、 The geometric meaning of double integral ：

Represents the volume of the curved top cylinder

2、 The properties of double integral ：

（1）、 $\iint_{D}^{}[\alpha f(x,y)+\beta g(x,y)]d\sigma =\alpha \iint_{D}^{}f(x,y)d\sigma +\beta \iint_{D}^{}g(x,y)d\sigma$

（2）、 $\iint_{D}^{}f(x,y)d\sigma =\iint_{D1}^{}f(x,y)d\sigma +\iint_{D2}^{}f(x,y)d\sigma$

（3）、 $\sigma =\iint_{D}^{}1\cdot d\sigma =\iint_{D}^{}d\sigma$

（4）、 If in D On $f(x,y)\leqslant g(x,y)$ , be $\iint_{D}^{}f(x,y)d\sigma \leqslant \iint_{D}^{}g(x,y)d\sigma$

（5）、M and m Respectively D The maximum and minimum values on , be $m\sigma \leqslant \iint_{D}^{}f(x,y)d\sigma \leqslant \iint_{D}^{}Md\sigma$

（6）、f（x,y） stay D Continuous on , $\sigma$ yes D Area on , It's in D There is at least one thing on the （ $\xi$ , $\eta$ ）, bring

$\iint_{D}^{}f(x,y)d\sigma =f(\xi ,\eta )\sigma$

（7）、 symmetry , Even times odd zero ,（D About y symmetry , see x;D About x symmetry , see y）

3、 Calculation of double integral

1） Calculation formula of double integral in rectangular coordinate system , Area micro element

Two calculation formulas can be obtained by converting it into repeated integral calculation ：

① First pair y integral , Right again x integral ：;

② First pair x integral , Right again y integral ：．

（2） Calculation formula of double integral in polar coordinate system , Area micro element , The conversion formula between rectangular coordinates and polar coordinates is ：．

Use polar coordinates to calculate double integral , First of all, we should distinguish the position relationship between the pole and the integral region , Then determine the integral limit , Divide the double product into successive integrals for calculation , There are three specific situations ：

① The pole is in the integral region D Outside ：;

② The pole is in the integral region D Inside ：;

③ The pole is in the integral region D On the border of ：

4、 Calculation of triple integral

（1）、 projection （ First one, then two ） Most use

$\iiint_{\Omega }^{}{}f(x,y,z)dv=\int_{a}^{b}dx\int_{y1(x)}^{y2(x)}dy\int_{z1(x,y)}^{z2(x,y)}f\left ( x,y,z \right )dz$

（2）、 Section method （ First two then one ） The integrand function does not contain x,y And when the street cross-sectional area is easy to find

$\iiint_{\Omega }^{}f(x,y,z)dv=\int_{c1}^{c2}dz\iint_{Dz}^{}f(x,y,z)dxdy$

（3）、 Use cylindrical coordinates to calculate Project as part of a circle , The equation is simple, that is, the variables can be separated （ Functions only have multiplication and division , No addition or subtraction ）

$\left\{\begin{matrix} x=\rho cos\theta \\ y=\rho sin\theta \\ z=z \end{matrix}\right.$ , $\iiint_{\Omega }^{}f(x,y,z)dxdydz=\iiint_{\Omega }^{}F(\rho ,\theta ,z)\rho d\rho d\theta dz$

*（4）、 Spherical coordinates The integrand function contains $x^{^{2}}$ , $y^{2}$ , $z^{2}$ Space is a ball or part of a ball

$\left\{\begin{matrix} x=rsin\varphi cos\theta \\ y=rsin\varphi sin\theta \\ z=rcos\varphi \end{matrix}\right.$

$\iiint_{\Omega }^{}f(x,y,z)dxdydz=\iiint_{\Omega }^{}F(r,\varphi ,\theta )r^{2 }sin\varphi drd\varphi d\theta =\int_{0}^{2\pi }d\theta \int_{0}^{\pi }d\varphi \int_{0}^{r(\varphi ,\theta )}F(r,\varphi ,\theta )r^{2}sin\varphi dr$

5、 Calculate the surface area by multiple integral

$A=\iint_{D}^{}\sqrt{1+fx^{2+fy^{2}}}d\sigma$

The first 11 Chapter Curve integral and surface integral

1. The curve integral of the arc length

L The parameter equation of is $\left\{\begin{matrix} x=\varphi (t)\\ y=\psi (t) \end{matrix}\right.$ , $(\alpha \leq t \leq \beta )$

$\int_{L}^{}f(x,y)ds=\int_{\alpha }^{\beta }f[\varphi (t),\psi (t)]$ $\sqrt{{\varphi }'^{2}(t)+{\psi }'^{2}(t)}$ （ $\alpha < \beta$ ）

Always greater than or equal to zero

2、 Curvilinear integration of coordinates

$\int_{L}^{}P(x,y)dx+Q(x,y)dy=$ $\int_{\alpha }^{\beta }\left \{ P[\varphi (t),\psi (t)]{\varphi }'(t)+Q[\varphi (t),\psi (t)]{\psi }'(t) \right \}dt$

α As a starting point ,β End point ,α It doesn't have to be less than β

3、 The connection between two kinds of curve integrals

$\int_{L}^{}Pdx+Qdy=\int_{L}^{}(Pcos\alpha +Qcos\beta )ds$ ,

among $cos\alpha =\frac{{\varphi }'(t)}{\sqrt{{\varphi }'^{2}(t)+{\psi }'^{2}(t)}}$ , $cos\beta =\frac{{\psi }'(t)}{\sqrt{{\varphi }'^{2}(t)+{\psi }'^{2}(t)}}$

4、 Green's formula

Closed area D from Piecewise smooth The curve of L Surround ,P,Q stay D There is a continuous partial derivative of the first order , be

$\iint_{D}^{}\left ( \frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y}\right )dxdy=\oint_{L}^{}Pdx+Qdy$ , among L yes D take positive （ Anti-clockwise ） Boundary curve of

5、 The condition that the curve integral on the plane is independent of the path

The curve integral is independent of the path $\Leftrightarrow \frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$

*6、 Use curve integral to express the area of plane graph

$s=\frac{1}{2}\oint_{L}^{}xdy-ydx$

7、 Surface integral over area

$\iint_{\Sigma }^{}f(x,y,z)ds=$ $\iint_{Dxy}^{}f[x,y,z(x,y)]$ $\sqrt{1+z_{x}^{2}(x,y)+z_{y}^{2}(x,y)}dxdy$

Independent of the direction of the surface

One investment, two generations and three micro changes

There is parity （Σ About xoy symmetry , see f About z Odd and even , Even times odd zero ）, symmetry

8、 Surface integral of coordinates

$\iint_{\Sigma }^{}R(x,y,z)dxdy=\iint_{D_{xy}}^{}R[x,y,z(x,y)]dxdy$

One vote, two generations and three fixed numbers

9、 The connection between the two types of curved area fractions

$\iint_{\Sigma }^{}Pdydz+Qdzdx+Rdxdy=\iint_{\Sigma }^{}(Pcos\alpha +Qcos\beta +Rcos\gamma )ds$ , among cosα,cosβ,cosγ by Σ stay （x,y,z） The direction cosine of the normal vector at

10、 Gauss formula

$\iiint_{\Omega }^{}(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z})dv$ $=\oiint_{\Sigma }^{}Pdydz+Qdzdx+Rdxdy$