{"id":752,"date":"2022-07-02T09:07:48","date_gmt":"2022-07-02T01:07:48","guid":{"rendered":"https:\/\/damakey.com\/?p=752"},"modified":"2023-05-24T16:55:03","modified_gmt":"2023-05-24T08:55:03","slug":"%e5%b0%8d%e6%95%b8%e8%9e%ba%e7%b7%9a%e7%ad%89%e8%a7%92%e8%9e%ba%e7%b7%9a","status":"publish","type":"post","link":"https:\/\/damakey.com\/?p=752","title":{"rendered":"\u5c0d\u6578\u87ba\u7dda(\u7b49\u89d2\u87ba\u7dda)"},"content":{"rendered":"

<\/a>
<\/span><\/p>\n

 <\/p>\n

\u984c\u76ee\uff1a\u5728\u4e00\u500b\u908a\u9577\u70ba<\/span>a<\/span>\u7684\u6b63\u65b9\u5f62\u7684\u56db\u500b\u908a\u89d2\u4e0a\u5404\u6709\u4e00\u96bb\u72d7\uff0c\u56db\u96bb\u72d7\u6309\u8457\u9006\u6642\u91dd\u7684\u65b9\u5411\u8ffd\u9010\u81e8\u8fd1\u7684\u90a3\u96bb\u72d7\u3002\u5982\u679c\u7260\u5011\u540c\u6642\u51fa\u767c\uff0c\u4ee5\u540c\u6a23\u7684\u901f\u7387\u524d\u9032\uff0c\u90a3\u9ebc\u5728\u6b63\u65b9\u5f62\u7684\u4e2d\u592e\u6703\u5408\u6642\uff0c\u6bcf\u96bb\u72d7\u5404\u8dd1\u4e86\u591a\u5c11\u8ddd\u96e2\uff1f<\/span>*<\/span><\/p>\n

 <\/p>\n

\u7a0d\u5fae\u60f3\u50cf\u4e00\u4e0b\uff0c\u6bcf\u96bb\u72d7\u8d70\u7684\u8def\u7dda\uff0c\u61c9\u8a72\u662f\u4e00\u689d\u5411\u6b63\u65b9\u5f62\u4e2d\u5fc3\u9010\u6f38\u6536\u6582\u7684\u7dda\u3002<\/span><\/p>\n

 <\/p>\n

\u6211\u5011\u53ef\u4ee5\u7528\u6975\u5ea7\u6a19\u4f86\u8868\u793a\u548c\u6c42\u89e3\u3002<\/span><\/p>\n

 <\/p>\n

\u4e00\u958b\u59cb\u7684\u6642\u5019\uff0c\u5047\u8a2d\u72d7\u96e2\u6b63\u65b9\u5f62\u4e2d\u5fc3\u7684\u8ddd\u96e2\u662f<\/span>r<\/span>\u3002\u5982\u679c\u72d7\u671d\u8457\u4ee5<\/span>r<\/span>\u70ba\u534a\u5f91\u7684\u5713\u8dd1\uff0c\u5728\u8ffd\u9010\u7684\u67d0\u500b\u77ed\u66ab\u6642\u9593<\/span> dt(delta t)<\/span>\uff0c\u72d7\u504f\u96e2\u6b63\u65b9\u5f62\u4e2d\u5fc3\u9ede\u7684\u89d2\u5ea6\u70ba<\/span>dθ(delta θ)<\/span>\uff0c\u90a3\u72d7\u8dd1\u7684\u8ddd\u96e2\u53ef\u7528<\/span>rdθ<\/span>\u4f30\u8a08<\/span>(<\/span>\u4e00\u6bb5<\/span>dθ<\/span>\u7684\u5f27\u9577<\/span>)<\/span>\u3002<\/span><\/p>\n

 <\/p>\n

\u7136\u800c\uff0c\u56e0\u70ba\u72d7\u662f\u7784\u6e96\u4e0b\u4e00\u96bb\u72d7\u5728\u8dd1\uff0c\u5728<\/span>dt<\/span>\u6642\uff0c\u5047\u8a2d\u72d7\u8dd1\u4e86<\/span>ds (delta s)<\/span>\uff0c\u72d7\u5728\u65b0\u7684\u4f4d\u7f6e\u8ddd\u6b63\u65b9\u5f62\u4e2d\u5fc3\u6703\u66f4\u9760\u8fd1\u4e00\u9ede\u9ede\uff0c\u8b6c\u5982\u5047\u8a2d\u662f<\/span>dr(delta r)<\/span>\u3002<\/span><\/p>\n

 <\/p>\n

\u5728\u8ffd\u9010\u7684\u67d0\u500b\u77ed\u66ab\u6642\u9593<\/span> dt(delta t)<\/span>\uff0c\u72d7\u96e2\u6b63\u65b9\u5f62\u4e2d\u5fc3\u7684\u89d2\u5ea6\u70ba<\/span>dθ(delta θ)<\/span>\uff0c\u72d7\u8dd1\u7684\u8ddd\u96e2\u5047\u8a2d<\/span>ds<\/span>\u3002\u53ef\u7528<\/span>rdθ<\/span>\u4f30\u8a08<\/span>(<\/span>\u4e00\u6bb5<\/span>dθ<\/span>\u7684\u5f27\u9577<\/span>)<\/span>\uff0c\u800c\u72d7\u5411\u4e0b\u4e00\u96bb\u8d70\u8ffd\u9010<\/span>dt<\/span>\u6642\u9593\u5230\u9054\u7684\u65b0\u4f4d\u7f6e\uff0c\u5047\u8a2d\u5f80\u6b63\u65b9\u5f62\u4e2d\u5fc3\u63a5\u8fd1\u4e86<\/span>dr(<\/span>\u56e0\u70ba\u534a\u5f91\u8b8a\u6210\u66f4\u77ed\u7684<\/span>r-dr<\/span>\uff0c\u6240\u4ee5\u6975\u5ea7\u6a19\u8a18\u6210<\/span>-dr<\/span>\uff0c\u56e0\u70ba\u66f4\u8fd1\u539f\u9ede\u4e86\u3002<\/span><\/p>\n

 <\/p>\n

\u9019\u6642\u53ef\u4ee5\u89c0\u5bdf\u5230\uff0c\u96a8\u8457<\/span>dt <\/span>\u8b8a\u5f97\u7121\u9650\u5c0f\uff0c<\/span>ds <\/span>\u53ef\u8996\u70ba\u4e00\u500b\u7121\u9650\u5c0f\u7b49\u8170\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u659c\u908a\uff0c\u5b83\u53e6\u5916\u5169\u500b\u908a\u5206\u5225\u70ba<\/span>rdθ <\/span>\u548c<\/span>-dr <\/span>\u3002<\/span><\/p>\n

 <\/p>\n

\u56e0\u70ba\u9019\u500b\u7b49\u8170\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u5169\u500b\u5e95\u89d2\u70ba<\/span>\ud835\udf45<\/span>\/4(45<\/span>\u5ea6\u89d2<\/span>)<\/span>\uff0c\u6240\u4ee5\uff1a<\/span><\/p>\n

 <\/p>\n

Tan(<\/span>\ud835\udf45<\/span>\/4)=-dr\/rdθ=1<\/span><\/p>\n

 <\/p>\n

dr\/r=-dθ<\/span><\/p>\n

 <\/p>\n

\u5169\u908a\u7a4d\u5206\uff0c\u5f97\u5230<\/span><\/p>\n

 <\/p>\n

Lnr =-θ+C <\/span><\/p>\n

 <\/p>\n

\u6216\u5beb\u6210\u6307\u6578\u51fd\u6578<\/span><\/p>\n

 <\/p>\n

 r(θ)=Ke^(-θ), <\/span>\u5176\u4e2d\u5e38\u6578<\/span>K=e^C<\/span><\/p>\n

 <\/p>\n

\u9019\u5c31\u767d\u52aa\u5229<\/span>(Bernoulli)<\/span>\u5144\u5f1f\u7387\u5148\u7814\u7a76\u7684\u300c\u5c0d\u6578\u87ba\u65cb\u300d<\/span>(logarithmic spiral)<\/span>\u3002<\/span><\/p>\n

 <\/p>\n

\u5982\u679c\u6211\u5011\u5047\u8a2d\u56db\u96bb\u72d7\u7684\u8d77\u59cb\u4f4d\u7f6e\uff0c\u5728\u76f4\u89d2\u5ea7\u6a19\u7684\u8ef8\u7dda\u4e0a\uff0c\u6bcf\u96bb\u72d7\u76f8\u8ddd<\/span>a<\/span>\u5c31\u662f\u6bcf\u5169\u96bb\u72d7\u8207\u539f\u9ede\u6240\u5f62\u6210\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u659c\u908a\uff0c\u6240\u4ee5\u7b2c\u4e00\u96bb\u72d7\u7684\u8d77\u59cb\u4f4d\u7f6e\u5c31\u5728<\/span>(a\/2^(1\/2)<\/span>\uff0c<\/span>0<\/span>\uff09\uff0c<\/span>a\/2^(1\/2)<\/span>\u5c31\u662f\u7260\u548c\u539f\u9ede\u7684\u8ddd\u96e2\u3002<\/span><\/p>\n

 <\/p>\n

r(θ=0)= a\/2^(1\/2)=K<\/span><\/p>\n

 <\/p>\n

\u628a<\/span>K<\/span>\u7528<\/span> a\/2^(1\/2)<\/span>\u53d6\u4ee3<\/span><\/p>\n

 <\/p>\n

r(θ)= a\/(2^(1\/2) )*e^(-θ) <\/span>\u9019\u5c31\u662f\u7b2c\u4e00\u96bb\u72d7\u7528\u6975\u5ea7\u6a19\u8868\u793a\u6240\u8dd1\u7684\u8ecc\u8de1\u3002<\/span><\/p>\n

 <\/p>\n

\u90a3\u9019\u96bb\u72d7\u8dd1\u5230\u539f\u9ede\uff0c\u5171\u8dd1\u591a\u9060\u5462\uff1f\u6982\u5ff5\u5c31\u662f\u5f9e<\/span>θ=0<\/span>\u7e5e\u8457\u539f\u9ede\u5230<\/span>θ= ∞<\/span>\uff0c\u628a\u6bcf\u6bb5<\/span>ds <\/span>\u7d2f\u7a4d\u8d77\u4f86\u5c31\u53ef\u4ee5<\/span>(<\/span>\u7a4d\u5206<\/span>)<\/span><\/p>\n

 <\/p>\n

\u7531\u7562\u6c0f\u5b9a\u7406\uff0c\u72d7\u5728<\/span>dt <\/span>\u6642\u9593\u5167\u6240\u8dd1\u7684\u8ddd\u96e2<\/span>ds <\/span>\u662f\u524d\u8ff0\u90a3\u500b\u7121\u9650\u5c0f\u76f4\u89d2\u7b49\u8170\u4e09\u89d2\u5f62\u7684\u659c\u908a\uff0c\u8170\u908a\u662f<\/span>rdθ<\/span>\uff0c\u90a3\u9ebc<\/span><\/p>\n

 <\/p>\n

ds=2^(1\/2)* rdθ= 2^(1\/2)* a\/(2^(1\/2) )e^(-θ)dθ =ae^(-θ) dθ<\/span><\/p>\n

 <\/p>\n

\u72d7\u8dd1\u5230\u539f\u9ede\u7684\u7e3d\u8ddd\u96e2<\/span> ∫ ds =a ∫ e^(-θ) dθ=a(-e^(-θ)) (θ<\/span>\u7531<\/span>0<\/span>\u5230<\/span>∞)=-a(0-1)=a<\/span><\/p>\n

 <\/p>\n

\u5728\u72d7\u65bc\u63a5\u8fd1\u539f\u9ede\u76f8\u6703\u7684\u6642\u5019\uff0c\u5c45\u7136\u90fd\u6b63\u597d\u8d70\u4e86<\/span>a<\/span>\u90a3\u9ebc\u9577\u7684\u8ddd\u96e2\uff0c\u7b49\u65bc\u7260\u4e4b\u9593\u4e00\u958b\u59cb\u7684\u8ddd\u96e2\uff01<\/span><\/p>\n

 <\/p>\n

\u5728\u76f4\u89c0\u4e0a\uff0c\u600e\u9ebc\u89e3\u91cb\u9019\u500b\u5947\u5999\u7684\u8ddd\u96e2<\/span>a<\/span>\u5462\uff1f\u4f55\u82e6\u4e92\u76f8\u8ffd\u9010\uff0c\u4e0b\u4e00\u96bb\u72d7\u7b49\u8457\u524d\u4e00\u96bb\u72d7\u8d70\u76f4\u7dda\u4e0d\u5c31\u5f97\u4e86\uff0c\u78ba\u5be6\u3002<\/span><\/p>\n

 <\/p>\n

\u5982\u679c\u6211\u5011\u5728\u72d7\u7684\u982d\u4e0a\u88dd\u651d\u5f71\u6a5f\uff0c\u5982\u679c\u72d7\u4e00\u76f4\u4fdd\u6301\u5c0d\u8457\u4e0b\u4e00\u96bb\u72d7\uff0c\u90a3\u9ebc\u5f71\u50cf\u4e2d\u5c31\u6703\u770b\u5230\u4e0b\u4e00\u96bb\u72d7\u4e00\u76f4\u5728\u5f71\u50cf\u7684\u4e2d\u9593\uff0c\u8ffd\u9010\u7684\u72d7\u5f77\u4f5b\u662f\u76f4\u76f4\u8d70\u4e86<\/span>a<\/span>\u9019\u500b\u8ddd\u96e2\u800c\u5df2\uff0c\u70ba\u4ec0\u9ebc\u5462\uff1f\u56e0\u70ba\u4e0b\u4e00\u96bb\u72d7\u7684\u504f\u5411\u7684\u5206\u5411\u91cf\uff0c\u88ab\u5f8c\u4e00\u96bb\u72d7\u4e0d\u65b7\u5c0d\u6e96\u8f49\u5411\u7684\u5206\u5411\u91cf\u300c\u62b5\u6d88\u300d\u4e86\uff01<\/span><\/p>\n

 <\/p>\n

\u524d\u8ff0\u7684\u4f8b\u5b50\u4ee5\u6bcf\u96bb\u72d7\u5c0d\u6e96\u4e0b\u4e00\u96bb\u72d7\uff0c\u56fa\u5b9a\u593e\u89d2\u70ba<\/span>\ud835\udf45<\/span>\/4(45<\/span>\u5ea6\u89d2<\/span>)<\/span>\u6240\u4ee5\u7528<\/span>Tan(<\/span>\ud835\udf45<\/span>\/4)=1 <\/span>\u4ee3\u5165\u3002\u800c\u5982\u679c\u593e\u89d2\u662f\u53e6\u5916\u4e00\u500b\u503c<\/span>a<\/span>\uff0c<\/span>0<a<<\/span>\ud835\udf45<\/span>\uff0c<\/span>a≠<\/span>\ud835\udf45<\/span>\/2<\/span>\uff0c\u90a3\u9ebc\u5c31\u662f\u7528<\/span>Tan(a)<\/span>\u4ee3\u5165<\/span><\/p>\n

 <\/p>\n

Tan(a)=-dr\/rdθ<\/span><\/p>\n

 <\/p>\n

dr\/r=-tan(a)dθ<\/span><\/p>\n

 <\/p>\n

\u5169\u908a\u7a4d\u5206\uff0c\u5f97\u5230<\/span><\/p>\n

 <\/p>\n

Lnr =-tan(a)*θ+C , C<\/span>\u70ba\u5e38\u6578<\/span><\/p>\n

 <\/p>\n

\u6216\u5beb\u6210\u6307\u6578\u51fd\u6578<\/span><\/p>\n

 <\/p>\n

r(θ)=e^(-tan(a)*θ+C)= e^C *e^(-tan(a)*θ)<\/span><\/p>\n

 <\/p>\n

r(θ)<\/span>\u96a8\u8457<\/span>θ<\/span>\u7684\u589e\u52a0\u800c\u5448\u73fe\u6307\u6578\u6027\u8cea\u7684\u52a0\u901f\u7e2e\u77ed\uff0c\u662f\u4e00\u689d\u5411\u5167\u6372\u7684\u7b49\u89d2\u87ba\u7dda\uff0c\u7b49\u89d2\u70ba<\/span>a(<\/span>\u6bcf\u500b\u77ac\u9593\u5411\u5167\u7684\u5207\u89d2\u70ba<\/span>a)<\/span>\u3002<\/span><\/p>\n

 <\/p>\n

a=0, tan(a)=0, <\/span>\u96a8\u8457<\/span>a<\/span>\u589e\u52a0<\/span>tan(a)<\/span>\u5728<\/span>a= <\/span>\ud835\udf45<\/span>\/4<\/span>\u4e4b\u5f8c\u5448\u73fe\u6307\u6578\u578b\u614b\u7684\u8fc5\u901f\u589e\u52a0\uff0c\u5728<\/span>a= <\/span>\ud835\udf45<\/span>\/2<\/span>\u9644\u8fd1\u63a5\u8fd1\u7121\u9650\u5927\uff0c\u53ef\u898b\u66f4\u5927\u7684\u7b49\u89d2<\/span>a<\/span>\uff0c\u6703\u4f7f\u5f97<\/span>r(θ)<\/span>\u7684\u6536\u6582\u52a0\u901f\uff0c\u9019\u6703\u4f7f\u5f97\u87ba\u7dda\u7684\u5f62\u72c0\u770b\u8d77\u4f86\u6bd4\u8f03\u7626\u9661\u3002\u800c\u8f03\u5c0f\u7b49\u89d2\u7684\u87ba\u7dda\uff0c\u5247\u662f\u6162\u6162\u5411\u539f\u9ede\u6536\u6582\uff0c\u87ba\u7dda\u6703\u986f\u5f97\u6bd4\u8f03\u5e73\u7de9\uff0c\u7e5e\u5708\u4e5f\u6703\u6bd4\u8f03\u5bc6\u3002<\/span><\/p>\n

 <\/p>\n

\u7b49\u89d2<\/span>a<\/span>\u5728\u4ecb\u65bc<\/span>\ud835\udf45<\/span>\/2<\/span>\u5230<\/span>\ud835\udf45<\/span>\u4e4b\u9593\uff0c<\/span>tan(a)<\/span>\u7531\u8ca0\u7684\u7121\u9650\u5c0f\u589e\u52a0\u5230<\/span>0<\/span>\uff0c\u87ba\u7dda\u5c31\u7531\u5e7e\u4e4e\u7121\u9650\u5927<\/span>(<\/span>\u7121\u9650\u9060\u7684\u5730\u65b9<\/span>)<\/span>\u5411\u539f\u9ede\u7e5e\uff0c\u4f46\u6703\u505c\u5728<\/span>e^C <\/span>\u9019\u500b\u9577\u5ea6\u4e0a\uff0c\u4e0d\u6703\u7d93\u904e\u539f\u9ede\u3002<\/span><\/p>\n

 <\/p>\n

\u5f88\u591a\u751f\u7269\u7684\u751f\u9577\u8ecc\u8de1\uff0c\u8b6c\u5982\u5411\u65e5\u8475\u7684\u82b1\u3001\u9cf3\u68a8\u4e0a\u9762\u7684\u65cb\uff0c\u9e1a\u9d61\u87ba\u7684\u6bbb\u7b49\u7b49\uff0c\u5927\u81f4\u90fd\u662f\u7b49\u89d2\u87ba\u7dda\uff0c\u70ba\u4ec0\u9ebc\u5462\uff1f\u56e0\u70ba\u5b83\u5011\u7684\u751f\u9577\uff0c\u90fd\u662f\u5faa\u5e8f\u6f38\u9032\uff0c\u800c\u4e14\u5448\u73fe\u7b49\u89d2\u7684\u767c\u5c55\u7684\u3002\u5b83\u5011\u4e0d\u662f\u6578\u5b78\u5bb6\uff0c\u800c\u662f\u7b49\u89d2\u87ba\u7dda\u7684\u6027\u8cea\uff0c\u525b\u597d\u53ef\u4ee5\u7528\u4f86\u63cf\u8ff0\u5b83\u5011\u7d93\u904e\u9032\u5316\u6dd8\u6c70\u8cfd\u8003\u9a57\u4e4b\u5f8c\uff0c\u6700\u6709\u6548\u80fd\u7684\u751f\u547d\u683c\u5f0f\u3002<\/span><\/p>\n

<\/span><\/p>\n

\u559c\u6b61\u5c0d\u6578\u87ba\u7dda<\/span>(<\/span>\u7b49\u89d2\u87ba\u7dda<\/span>)<\/span>\u7684\u7f8e\uff0c\u5c31\u662f\u6b61\u559c\u751f\u547d\u512a\u96c5\u7684\u7bc0\u594f\u3002<\/span><\/p>\n

 <\/p>\n

 <\/p>\n

*<\/span>\uff1a\u300a\u5b78\u5fae\u7a4d\u5206\uff0c\u4e5f\u5b78\u4eba\u751f\u300b\uff0c\u53f2\u8482\u82ac·\u53f2\u7279\u683c\u8332<\/span> <\/span>\u8457\uff0c\u8521\u627f\u5fd7<\/span> <\/span>\u8b6f<\/span><\/p>\n

 <\/p>\n

2022\/7\/1 <\/span>\u5c0d\u6578\u87ba\u7dda<\/span>(<\/span>\u7b49\u89d2\u87ba\u7dda<\/span>) Damakey<\/span><\/p>\n

 <\/p>\n

<\/a>
<\/a>
<\/span><\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

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