Files
thejinchao.github.io/assets/math_03.html-or7U2Ler.js
T

2 lines
15 KiB
JavaScript
Raw Blame History

This file contains invisible Unicode characters
This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
import{_ as c,c as a,a as x,b as m,d as s,o as j}from"./app-DCgfI_Ar.js";const e="/assets/Steradian-cZYOZ3WD.svg",i="/assets/solid_angle-Clr9v3xM.svg",t={};function n(u,l){return j(),a("div",null,l[0]||(l[0]=[x('<h1 id="数学基础-三" tabindex="-1"><a class="header-anchor" href="#数学基础-三"><span>数学基础(三)</span></a></h1><hr><h2 id="_4-立体角" tabindex="-1"><a class="header-anchor" href="#_4-立体角"><span>4. 立体角</span></a></h2><h3 id="_4-1-定义" tabindex="-1"><a class="header-anchor" href="#_4-1-定义"><span>4.1 定义</span></a></h3>',4),m("p",null,[s("三维空间的弧度,以r为半径的球,在球表面面积为"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D434 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"A"),m("mo",null,"="),m("msup",null,[m("mi",null,"r"),m("mn",null,"2")])])])]),s("的圆对应的立体角为"),m("strong",null,"单位立体角"),s(",")],-1),m("figure",null,[m("img",{src:e,alt:"",tabindex:"0",loading:"lazy"}),m("figcaption")],-1),m("h3",{id:"_4-2-和平面角的对应关系",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_4-2-和平面角的对应关系"},[m("span",null,"4.2 和平面角的对应关系")])],-1),m("p",null,"下图中方形面积对应的立体角为",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D714 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mstyle",{space:"4"},[m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mrow",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D434 TEX-I"})])])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D445 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.289em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])])])])])])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mstyle",{space:"4"},[m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mrow",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D445 TEX-I"})]),m("mjx-mi",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c73"}),m("mjx-c",{class:"mjx-c69"}),m("mjx-c",{class:"mjx-c6E"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2061"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D445 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})])])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D445 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.289em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])])])])])])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c73"}),m("mjx-c",{class:"mjx-c69"}),m("mjx-c",{class:"mjx-c6E"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2061"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mi",null,"d"),m("mi",null,"ω"),m("mo",null,"="),m("mstyle",{displaystyle:"true",scriptlevel:"0"},[m("mfrac",null,[m("mrow",null,[m("mi",null,"d"),m("mi",null,"A")]),m("msup",null,[m("mi",null,"R"),m("mn",null,"2")])])]),m("mo",null,"="),m("mstyle",{displaystyle:"true",scriptlevel:"0"},[m("mfrac",null,[m("mrow",null,[m("mi",null,"R"),m("mi",null,"sin"),m("mo",{"data-mjx-texclass":"NONE"},"⁡"),m("mi",null,"θ"),m("mi",null,"d"),m("mi",null,"ϕ"),m("mo",null,"×"),m("mi",null,"R"),m("mi",null,"d"),m("mi",null,"θ")]),m("msup",null,[m("mi",null,"R"),m("mn",null,"2")])])]),m("mo",null,"="),m("mi",null,"sin"),m("mo",{"data-mjx-texclass":"NONE"},"⁡"),m("mi",null,"θ"),m("mi",null,"d"),m("mi",null,"ϕ"),m("mi",null,"d"),m("mi",null,"θ")])])],-1),m("figure",null,[m("img",{src:i,alt:"",tabindex:"0",loading:"lazy"}),m("figcaption")],-1),m("h3",{id:"_4-3-积分",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_4-3-积分"},[m("span",null,"4.3 积分")])],-1),m("p",null,"针对立体角的半球体积分可以转换为球面角",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msub",null,[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c222B TEX-S2"})]),m("mjx-script",{style:{"vertical-align":"-0.896em","margin-left":"-0.388em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c3A9"})])])])]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D453 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mover",null,[m("mjx-over",{style:{"padding-bottom":"0.105em","padding-left":"0.311em","margin-bottom":"-0.516em"}},[m("mjx-mo",{class:"mjx-n",style:{width:"0","margin-left":"-0.25em"}},[m("mjx-c",{class:"mjx-c20D7 TEX-V"})])]),m("mjx-base",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D714 TEX-I"})])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D714 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msubsup",{space:"4"},[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c222B TEX-S2"})]),m("mjx-script",{style:{"vertical-align":"-0.896em","margin-left":"-0.388em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.647em"}},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D70B TEX-I"})])]),m("mjx-spacer",{style:{"margin-top":"1.506em"}}),m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c30"})])])])]),m("mjx-msubsup",{space:"2"},[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c222B TEX-S2"})]),m("mjx-script",{style:{"vertical-align":"-0.896em","margin-left":"-0.388em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.647em"}},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D70B TEX-I"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2F"})])]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])]),m("mjx-spacer",{style:{"margin-top":"1.337em"}}),m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c30"})])])])]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D453 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mi",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c73"}),m("mjx-c",{class:"mjx-c69"}),m("mjx-c",{class:"mjx-c6E"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2061"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("msub",null,[m("mo",{"data-mjx-texclass":"OP"},"∫"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",{mathvariant:"normal"},"Ω")])]),m("mi",null,"f"),m("mo",{stretchy:"false"},"("),m("mrow",{"data-mjx-texclass":"ORD"},[m("mover",null,[m("mi",null,"ω"),m("mo",{stretchy:"false"},"→")])]),m("mo",{stretchy:"false"},")"),m("mi",null,"d"),m("mi",null,"ω"),m("mo",null,"="),m("msubsup",null,[m("mo",{"data-mjx-texclass":"OP"},"∫"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"0")]),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"2"),m("mi",null,"π")])]),m("msubsup",null,[m("mo",{"data-mjx-texclass":"OP"},"∫"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"0")]),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"π"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"/")]),m("mn",null,"2")])]),m("mi",null,"f"),m("mo",{stretchy:"false"},"("),m("mi",null,"θ"),m("mo",null,","),m("mi",null,"ϕ"),m("mo",{stretchy:"false"},")"),m("mi",null,"sin"),m("mo",{"data-mjx-texclass":"NONE"},"⁡"),m("mi",null,"θ"),m("mi",null,"d"),m("mi",null,"θ"),m("mi",null,"d"),m("mi",null,"ϕ")])])],-1),m("p",null,"比如求半球体的面积",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msub",null,[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c222B TEX-S2"})]),m("mjx-script",{style:{"vertical-align":"-0.896em","margin-left":"-0.388em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c3A9"})])])])]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D714 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msubsup",{space:"4"},[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c222B TEX-S2"})]),m("mjx-script",{style:{"vertical-align":"-0.896em","margin-left":"-0.388em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.647em"}},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D70B TEX-I"})])]),m("mjx-spacer",{style:{"margin-top":"1.506em"}}),m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c30"})])])])]),m("mjx-msubsup",{space:"2"},[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c222B TEX-S2"})]),m("mjx-script",{style:{"vertical-align":"-0.896em","margin-left":"-0.388em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.647em"}},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D70B TEX-I"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2F"})])]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])]),m("mjx-spacer",{style:{"margin-top":"1.337em"}}),m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c30"})])])])]),m("mjx-mi",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c73"}),m("mjx-c",{class:"mjx-c69"}),m("mjx-c",{class:"mjx-c6E"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2061"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D703 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D70B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("msub",null,[m("mo",{"data-mjx-texclass":"OP"},"∫"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",{mathvariant:"normal"},"Ω")])]),m("mi",null,"d"),m("mi",null,"ω"),m("mo",null,"="),m("msubsup",null,[m("mo",{"data-mjx-texclass":"OP"},"∫"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"0")]),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"2"),m("mi",null,"π")])]),m("msubsup",null,[m("mo",{"data-mjx-texclass":"OP"},"∫"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"0")]),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"π"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"/")]),m("mn",null,"2")])]),m("mi",null,"sin"),m("mo",{"data-mjx-texclass":"NONE"},"⁡"),m("mi",null,"θ"),m("mi",null,"d"),m("mi",null,"θ"),m("mi",null,"d"),m("mi",null,"ϕ"),m("mo",null,"="),m("mn",null,"2"),m("mi",null,"π")])])],-1)]))}const r=c(t,[["render",n],["__file","math_03.html.vue"]]),p=JSON.parse('{"path":"/note/graphics/math/math_03.html","title":"数学基础(三)","lang":"zh-CN","frontmatter":{},"headers":[{"level":2,"title":"4. 立体角","slug":"_4-立体角","link":"#_4-立体角","children":[{"level":3,"title":"4.1 定义","slug":"_4-1-定义","link":"#_4-1-定义","children":[]},{"level":3,"title":"4.2 和平面角的对应关系","slug":"_4-2-和平面角的对应关系","link":"#_4-2-和平面角的对应关系","children":[]},{"level":3,"title":"4.3 积分","slug":"_4-3-积分","link":"#_4-3-积分","children":[]}]}],"git":{"updatedTime":1740235322000,"contributors":[{"name":"thejinchao","username":"thejinchao","email":"thejinchao@gmail.com","commits":1,"url":"https://github.com/thejinchao"}]},"filePathRelative":"note/graphics/math/math_03.md"}');export{r as comp,p as data};