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150 lines
412 KiB
150 lines
412 KiB
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Disjunction and conjunction",data:[{value:"\\bigvee"},{value:"\\bigvee_{a}^{b}"},{value:"\\bigwedge"},{value:"\\bigwedge_{a}^{b}"}]}]},{name:"\u62EC\u53F7\u53D6\u6574",value:"\\left [ \\left ( \\right ) \\right ] ",children:[{name:"\u62EC\u53F7 Brackets",data:[{value:"\\left ( \\right )"},{value:"\\left [ \\right ]"},{value:"\\left \\langle \\right \\rangle "},{value:"\\left | \\right | "},{value:"\\left \\lfloor \\right \\rfloor "},{value:"\\left \\lceil \\right \\rceil "}]}]}],x=[{name:"\u4EE3\u6570",value:"\\sqrt{a^2+b^2}",children:[{data:[{value:"\\left(x-1\\right)\\left(x+3\\right) "},{value:"\\sqrt{a^2+b^2}"},{value:"\\left ( \\frac{a}{b}\\right )^{n}= \\frac{a^{n}}{b^{n}}"},{value:"\\frac{a}{b}\\pm \\frac{c}{d}= \\frac{ad \\pm bc}{bd} "},{value:"\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 "},{value:"\\frac{1}{\\sqrt{a}}=\\frac{\\sqrt{a}}{a},a\\ge 0\\frac{1}{\\sqrt{a}}=\\frac{\\sqrt{a}}{a},a\\ge 0 "},{value:"\\sqrt[n]{a^{n}}=\\left ( \\sqrt[n]{a}\\right )^{n} "},{value:"x ={-b \\pm \\sqrt{b^2-4ac}\\over 2a} "},{value:"y-y_{1}=k \\left( x-x_{1}\\right) "},{value:`\\left\\{\\begin{matrix} \r
|
|
x=a + r\\text{cos}\\theta \\ \r
|
|
y=b + r\\text{sin}\\theta \r
|
|
\\end{matrix}\\right. `},{value:`\\begin{array}{l} \r
|
|
\\text{\u5BF9\u4E8E\u65B9\u7A0B\u5F62\u5982\uFF1A}x^{3}-1=0 \\ \r
|
|
\\text{\u8BBE}\\text{:}\\omega =\\frac{-1+\\sqrt{3}i}{2} \\ \r
|
|
x_{1}=1,x_{2}= \\omega =\\frac{-1+\\sqrt{3}i}{2} \\ \r
|
|
x_{3}= \\omega ^{2}=\\frac{-1-\\sqrt{3}i}{2} \r
|
|
\\end{array} `},{value:`\\begin{array}{l} \r
|
|
a\\mathop{{x}}\\nolimits^{{2}}+bx+c=0 \\ \r
|
|
\\Delta =\\mathop{{b}}\\nolimits^{{2}}-4ac \\ \r
|
|
\\left\\{\\begin{matrix} \r
|
|
\\Delta \\gt 0\\text{\u65B9\u7A0B\u6709\u4E24\u4E2A\u4E0D\u76F8\u7B49\u7684\u5B9E\u6839} \\ \r
|
|
\\Delta = 0\\text{\u65B9\u7A0B\u6709\u4E24\u4E2A\u76F8\u7B49\u7684\u5B9E\u6839} \\ \r
|
|
\\Delta \\lt 0\\text{\u65B9\u7A0B\u65E0\u5B9E\u6839} \r
|
|
\\end{matrix}\\right. \r
|
|
\\end{array} `},{value:`\\begin{array}{l} \r
|
|
a\\mathop{{x}}\\nolimits^{{2}}+bx+c=0 \\ \r
|
|
\\Delta =\\mathop{{b}}\\nolimits^{{2}}-4ac \\ \r
|
|
\\mathop{{x}}\\nolimits_{{1,2}}=\\frac{{-b \\pm \r
|
|
\\sqrt{{\\mathop{{b}}\\nolimits^{{2}}-4ac}}}}{{2a}} \\ \r
|
|
\\mathop{{x}}\\nolimits_{{1}}+\\mathop{{x}}\\nolimits_{{2}}=-\\frac{{b}}{{a}} \\ \r
|
|
\\mathop{{x}}\\nolimits_{{1}}\\mathop{{x}}\\nolimits_{{2}}=\\frac{{c}}{{a}} \r
|
|
\\end{array} `}]}]},{name:"\u51E0\u4F55",value:"\\Delta A B C ",children:[{data:[{value:"\\Delta A B C "},{value:"a \\parallel c,b \\parallel c \\Rightarrow a \\parallel b "},{value:"l \\perp \\beta ,l \\subset \\alpha \\Rightarrow \\alpha \\perp \\beta"},{value:`\\left.\\begin{matrix} \r
|
|
a \\perp \\alpha \\ \r
|
|
b \\perp \\alpha \r
|
|
\\end{matrix}\\right\\}\\Rightarrow a \\parallel b`},{value:"P \\in \\alpha ,P \\in \\beta , \\alpha \\cap \\beta =l \\Rightarrow P \\in l "},{value:`\\alpha \\perp \\beta , \\alpha \\cap \\beta =l,a \\subset \\alpha ,a \\perp l \r
|
|
\\Rightarrow a \\perp \\beta `},{value:`\\left.\\begin{matrix} \r
|
|
a \\subset \\beta ,b \\subset \\beta ,a \\cap b=P \\ \r
|
|
a \\parallel \\partial ,b \\parallel \\partial \r
|
|
\\end{matrix}\\right\\}\\Rightarrow \\beta \\parallel \\alpha `},{value:"\\alpha \\parallel \\beta , \\gamma \\cap \\alpha =a, \\gamma \\cap \\beta =b \\Rightarrow a \\parallel b "},{value:"A \\in l,B \\in l,A \\in \\alpha ,B \\in \\alpha \\Rightarrow l \\subset \\alpha "},{value:`\\left.\\begin{matrix} \r
|
|
m \\subset \\alpha ,n \\subset \\alpha ,m \\cap n=P \\ \r
|
|
a \\perp m,a \\perp n \r
|
|
\\end{matrix}\\right\\}\\Rightarrow a \\perp \\alpha `},{value:`\\begin{array}{c} \r
|
|
\\text{\u76F4\u89D2\u4E09\u89D2\u5F62\u4E2D,\u76F4\u89D2\u8FB9\u957Fa,b,\u659C\u8FB9\u8FB9\u957Fc} \\ \r
|
|
a^{2}+b^{2}=c^{2} \r
|
|
\\end{array}`}]}]},{name:"\u4E0D\u7B49\u5F0F",value:"a > b",children:[{data:[{value:"a > b,b > c \\Rightarrow a > c "},{value:"a > b,c > d \\Rightarrow a+c > b+d "},{value:"a > b > 0,c > d > 0 \\Rightarrow ac bd "},{value:`\\begin{array}{c} \r
|
|
a \\gt b,c \\gt 0 \\Rightarrow ac \\gt bc \\ \r
|
|
a \\gt b,c \\lt 0 \\Rightarrow ac \\lt bc \r
|
|
\\end{array}`},{value:"\\left | a-b \\right | \\geqslant \\left | a \\right | -\\left | b \\right | "},{value:"-\\left | a \\right |\\leq a\\leqslant \\left | a \\right | "},{value:"\\left | a \\right |\\leqslant b \\Rightarrow -b \\leqslant a \\leqslant \\left | b \\right | "},{value:"\\left | a+b \\right | \\leqslant \\left | a \\right | + \\left | b \\right | "},{value:`\\begin{array}{c} \r
|
|
a \\gt b \\gt 0,n \\in N^{\\ast},n \\gt 1 \\ \r
|
|
\\Rightarrow a^{n}\\gt b^{n}, \\sqrt[n]{a}\\gt \\sqrt[n]{b} \r
|
|
\\end{array}`},{value:`\\left( \\sum_{k=1}^n a_k b_k \\right)^{\\!\\!2}\\leq \r
|
|
\\left( \\sum_{k=1}^n a_k^2 \\right) \\left( \\sum_{k=1}^n b_k^2 \\right) `},{value:`\\begin{array}{c} \r
|
|
a,b \\in R^{+} \\ \r
|
|
\\Rightarrow \\frac{a+b}{{2}}\\ge \\sqrt{ab} \\ \r
|
|
\\left( \\text{\u5F53\u4E14\u4EC5\u5F53}a=b\\text{\u65F6\u53D6\u201C}=\\text{\u201D\u53F7}\\right) \r
|
|
\\end{array}`},{value:`\\begin{array}{c} \r
|
|
a,b \\in R \\ \r
|
|
\\Rightarrow a^{2}+b^{2}\\gt 2ab \\ \r
|
|
\\left( \\text{\u5F53\u4E14\u4EC5\u5F53}a=b\\text{\u65F6\u53D6\u201C}=\\text{\u201D\u53F7}\\right) \r
|
|
\\end{array}`},{value:`\\begin{array}{c} \r
|
|
H_{n}=\\frac{n}{\\sum \\limits_{i=1}^{n}\\frac{1}{x_{i}}}= \\frac{n}{\\frac{1}{x_{1}}+ \\frac{1}{x_{2}}+ \\cdots + \\frac{1}{x_{n}}} \\ G_{n}=\\sqrt[n]{\\prod \\limits_{i=1}^{n}x_{i}}= \\sqrt[n]{x_{1}x_{2}\\cdots x_{n}} \\ A_{n}=\\frac{1}{n}\\sum \\limits_{i=1}^{n}x_{i}=\\frac{x_{1}+ x_{2}+ \\cdots + x_{n}}{n} \\ Q_{n}=\\sqrt{\\sum \\limits_{i=1}^{n}x_{i}^{2}}= \\sqrt{\\frac{x_{1}^{2}+ x_{2}^{2}+ \\cdots + x_{n}^{2}}{n}} \\ H_{n}\\leq G_{n}\\leq A_{n}\\leq Q_{n} \r
|
|
\\end{array}`}]}]},{name:"\u79EF\u5206",value:"\\frac{\\mathrm{d}\\partial}{\\partial x}",children:[{data:[{value:"\\frac{\\mathrm{d}}{\\mathrm{d}x}x^n=nx^{n-1} "},{value:"\\frac{\\mathrm{d}}{\\mathrm{d}x}e^{ax}=a\\,e^{ax} "},{value:"\\frac{\\mathrm{d}}{\\mathrm{d}x}\\ln(x)=\\frac{1}{x} "},{value:"\\frac{\\mathrm{d}}{\\mathrm{d}x}\\sin x=\\cos x "},{value:"\\frac{\\mathrm{d}}{\\mathrm{d}x}\\cos x=-\\sin x "},{value:"\\int k\\mathrm{d}x = kx+C "},{value:"\\frac{\\mathrm{d}}{\\mathrm{d}x}\\tan x=\\sec^2 x "},{value:"\\frac{\\mathrm{d}}{\\mathrm{d}x}\\cot x=-\\csc^2 x "},{value:"\\int \\frac{1}{x}\\mathrm{d}x= \\ln \\left| x \\right| +C "},{value:"\\int \\frac{1}{\\sqrt{1-x^{2}}}\\mathrm{d}x= \\arcsin x +C "},{value:"\\int \\frac{1}{1+x^{2}}\\mathrm{d}x= \\arctan x +C "},{value:"\\int u \\frac{\\mathrm{d}v}{\\mathrm{d}x}\\,\\mathrm{d}x=uv-\\int \\frac{\\mathrm{d}u}{\\mathrm{d}x}v\\,\\mathrm{d}x "},{value:"f(x) = \\int_{-\\infty}^\\infty \\hat f(x)\\xi\\,e^{2 \\pi i \\xi x} \\,\\mathrm{d}\\xi "},{value:"\\int x^{\\mu}\\mathrm{d}x=\\frac{x^{\\mu +1}}{\\mu +1}+C, \\left({\\mu \\neq -1}\\right) "}]}]},{name:"\u4E09\u89D2",value:"e^{i \\theta}",children:[{name:"\u6C42\u548C Summation",data:[{value:"e^{i \\theta} "},{value:"\\left(\\frac{\\pi}{2}-\\theta \\right ) "},{value:"\\text{sin}^{2}\\frac{\\alpha}{2}=\\frac{1- \\text{cos}\\alpha}{2} "},{value:"\\text{cos}^{2}\\frac{\\alpha}{2}=\\frac{1+ \\text{cos}\\alpha}{2} "},{value:"\\text{tan}\\frac{\\alpha}{2}=\\frac{\\text{sin}\\alpha}{1+ \\text{cos}\\alpha} "},{value:"\\sin \\alpha + \\sin \\beta =2 \\sin \\frac{\\alpha + \\beta}{2}\\cos \\frac{\\alpha - \\beta}{2} "},{value:"\\sin \\alpha - \\sin \\beta =2 \\cos \\frac{\\alpha + \\beta}{2}\\sin \\frac{\\alpha - \\beta}{2} "},{value:"\\cos \\alpha + \\cos \\beta =2 \\cos \\frac{\\alpha + \\beta}{2}\\cos \\frac{\\alpha - \\beta}{2} "},{value:"\\cos \\alpha - \\cos \\beta =-2\\sin \\frac{\\alpha + \\beta}{2}\\sin \\frac{\\alpha - \\beta}{2} "},{value:"a^{2}=b^{2}+c^{2}-2bc\\cos A "},{value:"\\frac{\\sin A}{a}=\\frac{\\sin B}{b}=\\frac{\\sin C}{c}=\\frac{1}{2R} "},{value:"\\sin \\left ( \\frac{\\pi}{2}-\\alpha \\right ) = \\cos \\alpha "},{value:"\\sin \\left ( \\frac{\\pi}{2}+\\alpha \\right ) = \\cos \\alpha "}]}]},{name:"\u7EDF\u8BA1",value:"C_{r}^{n}",children:[{data:[{value:"C_{r}^{n} "},{value:"\\frac{n!}{r!(n-r)!} "},{value:"\\sum_{i=1}^{n}{X_i} "},{value:"\\sum_{i=1}^{n}{X_i^2} "},{value:"X_1, \\cdots,X_n "},{value:"\\frac{x-\\mu}{\\sigma} "},{value:"\\sum_{i=1}^{n}{(X_i - \\overline{X})^2} "},{value:`\\begin{array}{c} \r
|
|
\\text{\u82E5}P \\left( AB \\right) =P \\left( A \\right) P \\left( B \\right) \\\\ \r
|
|
\\text{\u5219}P \\left( A \\left| B\\right. \\right) =P \\left({B}\\right) \r
|
|
\\end{array}`},{value:"P(E) ={n \\choose k}p^k (1-p)^{n-k} "},{value:"P \\left( A \\right) = \\lim \\limits_{n \\to \\infty}f_{n}\\left ( A \\right ) "},{value:"P \\left( \\bigcup \\limits_{i=1}^{+ \\infty}A_{i}\\right) = \\prod \\limits_{i=1}^{+ \\infty}P{\\left( A_{i}\\right)} "},{value:`\\begin{array}{c} \r
|
|
P \\left( \\emptyset \\right) =0 \\\\ \r
|
|
P \\left( S \\right) =1 \r
|
|
\\end{array}`},{value:`\\begin{array}{c} \r
|
|
\\forall A \\in S \\\\ \r
|
|
P \\left( A \\right) \\ge 0 \r
|
|
\\end{array}`},{value:"P \\left( \\bigcup \\limits_{i=1}^{n}A_{i}\\right) = \\prod \\limits_{i=1}^{n}P \\left( A_{i}\\right) "},{value:`\\begin{array}{c} \r
|
|
S= \\binom{N}{n},A_{k}=\\binom{M}{k}\\cdot \\binom{N-M}{n-k} \\\\ \r
|
|
P\\left ( A_{k}\\right ) = \\frac{\\binom{M}{k}\\cdot \\binom{N-M}{n-k}}{\\binom{N}{n}} \r
|
|
\\end{array}`},{value:`\\begin{array}{c} \r
|
|
P_{n}=n! \\\\ \r
|
|
A_{n}^{k}=\\frac{n!}{\\left( n-k \\left) !\\right. \\right.} \r
|
|
\\end{array}`}]}]}],b=ee=>{a.current.setValue(a.current.getValue()+" "+ee.value+" 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> ${ye}-submenu > ${ye}-submenu-title`]:{height:Pe,lineHeight:`${Pe}px`},[`${ye}-item-group-list ${ye}-submenu-title,
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${ye}-submenu-title`]:{paddingInlineEnd:It}}};var Be=be=>{const{componentCls:ye,iconCls:Pe,itemHeight:ke,colorTextLightSolid:He,dropdownWidth:ut,controlHeightLG:xt,motionDurationMid:bt,motionEaseOut:It,paddingXL:jt,itemMarginInline:wt,fontSizeLG:ht,motionDurationSlow:yt,paddingXS:Ut,boxShadowSecondary:mn,collapsedWidth:tn,collapsedIconSize:Kt}=be,pe={height:ke,lineHeight:`${ke}px`,listStylePosition:"inside",listStyleType:"disc"};return[{[ye]:{"&-inline, &-vertical":Object.assign({[`&${ye}-root`]:{boxShadow:"none"}},pt(be))},[`${ye}-submenu-popup`]:{[`${ye}-vertical`]:Object.assign(Object.assign({},pt(be)),{boxShadow:mn})}},{[`${ye}-submenu-popup ${ye}-vertical${ye}-sub`]:{minWidth:ut,maxHeight:`calc(100vh - ${xt*2.5}px)`,padding:"0",overflow:"hidden",borderInlineEnd:0,"&:not([class*='-active'])":{overflowX:"hidden",overflowY:"auto"}}},{[`${ye}-inline`]:{width:"100%",[`&${ye}-root`]:{[`${ye}-item, ${ye}-submenu-title`]:{display:"flex",alignItems:"center",transition:[`border-color ${yt}`,`background ${yt}`,`padding ${bt} ${It}`].join(","),[`> ${ye}-title-content`]:{flex:"auto",minWidth:0,overflow:"hidden",textOverflow:"ellipsis"},"> *":{flex:"none"}}},[`${ye}-sub${ye}-inline`]:{padding:0,border:0,borderRadius:0,boxShadow:"none",[`& > ${ye}-submenu > ${ye}-submenu-title`]:pe,[`& ${ye}-item-group-title`]:{paddingInlineStart:jt}},[`${ye}-item`]:pe}},{[`${ye}-inline-collapsed`]:{width:tn,[`&${ye}-root`]:{[`${ye}-item, ${ye}-submenu ${ye}-submenu-title`]:{[`> ${ye}-inline-collapsed-noicon`]:{fontSize:ht,textAlign:"center"}}},[`> ${ye}-item,
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> ${ye}-item-group > ${ye}-item-group-list > ${ye}-item,
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> ${ye}-item-group > ${ye}-item-group-list > ${ye}-submenu > ${ye}-submenu-title,
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> ${ye}-submenu > ${ye}-submenu-title`]:{insetInlineStart:0,paddingInline:`calc(50% - ${ht/2}px - ${wt}px)`,textOverflow:"clip",[`
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${ye}-submenu-arrow,
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${ye}-submenu-expand-icon
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`]:{opacity:0},[`${ye}-item-icon, ${Pe}`]:{margin:0,fontSize:Kt,lineHeight:`${ke}px`,"+ span":{display:"inline-block",opacity:0}}},[`${ye}-item-icon, ${Pe}`]:{display:"inline-block"},"&-tooltip":{pointerEvents:"none",[`${ye}-item-icon, ${Pe}`]:{display:"none"},"a, a:hover":{color:He}},[`${ye}-item-group-title`]:Object.assign(Object.assign({},Ce.vS),{paddingInline:Ut})}}]};const Ye=be=>{const{componentCls:ye,motionDurationSlow:Pe,motionDurationMid:ke,motionEaseInOut:He,motionEaseOut:ut,iconCls:xt,iconSize:bt,iconMarginInlineEnd:It}=be;return{[`${ye}-item, ${ye}-submenu-title`]:{position:"relative",display:"block",margin:0,whiteSpace:"nowrap",cursor:"pointer",transition:[`border-color ${Pe}`,`background ${Pe}`,`padding ${Pe} ${He}`].join(","),[`${ye}-item-icon, ${xt}`]:{minWidth:bt,fontSize:bt,transition:[`font-size ${ke} ${ut}`,`margin ${Pe} ${He}`,`color ${Pe}`].join(","),"+ span":{marginInlineStart:It,opacity:1,transition:[`opacity ${Pe} ${He}`,`margin ${Pe}`,`color 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