{"id":1420,"date":"2024-10-14T00:00:25","date_gmt":"2024-10-13T16:00:25","guid":{"rendered":"https:\/\/www.fanyamin.com\/wordpress\/?p=1420"},"modified":"2024-10-14T00:01:23","modified_gmt":"2024-10-13T16:01:23","slug":"%e7%94%a8-latex-%e6%9d%a5%e8%a1%a8%e7%a4%ba%e6%95%b0%e5%ad%a6%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/www.fanyamin.com\/wordpress\/?p=1420","title":{"rendered":"\u7528 latex \u6765\u8868\u793a\u6570\u5b66\u516c\u5f0f"},"content":{"rendered":"<p>Markdown\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528 LaTeX \u6765\u5199\u6570\u5b66\u516c\u5f0f\u3002\u53ea\u9700\u5728\u516c\u5f0f\u524d\u540e\u52a0\u4e0a\u7f8e\u5143\u7b26\u53f7 <code class=\"mathjax-inline language-mathjax\">...<\/code> \u6765\u8868\u793a\u884c\u5185\u516c\u5f0f\uff0c\u6216\u4f7f\u7528 <code class=\"katex-inline\">...<\/code> \u6765\u8868\u793a\u72ec\u7acb\u884c\u7684\u516c\u5f0f\u3002<\/p>\n<p>\u53c2\u89c1 <a href=\"https:\/\/jupyterbook.org\/en\/stable\/content\/math.html\">https:\/\/jupyterbook.org\/en\/stable\/content\/math.html<\/a><\/p>\n<p>\u4ee5\u4e0b\u662f\u4e00\u4e9b\u4e2d\u5b66\u6570\u5b66\u5e38\u7528\u7684\u516c\u5f0f\u793a\u4f8b\uff1a<\/p>\n<h3>1. <strong>\u4e8c\u6b21\u65b9\u7a0b\u6c42\u6839\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\nx = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}<br \/>\n$$<br \/>\n\u7528\u4e8e\u6c42\u89e3\u4e8c\u6b21\u65b9\u7a0b $$ax^2 + bx + c = 0$$<\/p>\n<h3>2. <strong>\u52fe\u80a1\u5b9a\u7406<\/strong><\/h3>\n<p>$$<br \/>\na^2 + b^2 = c^2<br \/>\n$$<br \/>\n\u5176\u4e2d ( a ) \u548c ( b ) \u662f\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u4e24\u6761\u76f4\u89d2\u8fb9\uff0c( c ) \u662f\u659c\u8fb9\u3002<\/p>\n<h3>3. <strong>\u4e58\u6cd5\u516c\u5f0f\uff08\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\uff09<\/strong><\/h3>\n<p>$$<br \/>\n(a + b)^2 = a^2 + 2ab + b^2<br \/>\n$$<br \/>\n$$<br \/>\n(a - b)^2 = a^2 - 2ab + b^2<br \/>\n$$<\/p>\n<h3>4. <strong>\u5e73\u65b9\u5dee\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\na^2 - b^2 = (a - b)(a + b)<br \/>\n$$<\/p>\n<h3>5. <strong>\u4e00\u5143\u4e00\u6b21\u65b9\u7a0b<\/strong><\/h3>\n<p>$$<br \/>\nax + b = 0 \\quad \\Rightarrow \\quad x = -\\frac{b}{a}<br \/>\n$$<\/p>\n<h3>6. <strong>\u7b49\u5dee\u6570\u5217\u7684\u901a\u9879\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\na_n = a_1 + (n - 1) d<br \/>\n$$<br \/>\n\u5176\u4e2d ( a_1 ) \u662f\u9996\u9879\uff0c( d ) \u662f\u516c\u5dee\uff0c( n ) \u662f\u9879\u6570\u3002<\/p>\n<h3>7. <strong>\u7b49\u6bd4\u6570\u5217\u7684\u901a\u9879\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\na_n = a_1 \\cdot r^{n-1}<br \/>\n$$<br \/>\n\u5176\u4e2d $$ a_1 $$ \u662f\u9996\u9879\uff0c$$ r $$ \u662f\u516c\u6bd4\uff0c$$ n $$ \u662f\u9879\u6570\u3002<\/p>\n<h3>8. <strong>\u5706\u7684\u9762\u79ef\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\nA = \\pi r^2<br \/>\n$$<br \/>\n\u5176\u4e2d ( r ) \u662f\u5706\u7684\u534a\u5f84\u3002<\/p>\n<h3>9. <strong>\u5706\u7684\u5468\u957f\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\nC = 2 \\pi r<br \/>\n$$<br \/>\n\u5176\u4e2d ( r ) \u662f\u5706\u7684\u534a\u5f84\u3002<\/p>\n<h3>10. <strong>\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u3001\u6b63\u5207<\/strong><\/h3>\n<p>$$<br \/>\n\\sin \\theta = \\frac{\\text{\u5bf9\u8fb9}}{\\text{\u659c\u8fb9}}<br \/>\n\\quad \\cos \\theta = \\frac{\\text{\u90bb\u8fb9}}{\\text{\u659c\u8fb9}}<br \/>\n\\quad \\tan \\theta = \\frac{\\text{\u5bf9\u8fb9}}{\\text{\u90bb\u8fb9}}<br \/>\n$$<\/p>\n<p>\u4ee5\u4e0b\u662f\u66f4\u591a\u7684\u5e38\u7528\u6570\u5b66\u516c\u5f0f\uff0c\u6db5\u76d6\u4ee3\u6570\u3001\u51e0\u4f55\u3001\u4e09\u89d2\u3001\u5fae\u79ef\u5206\u7b49\u9886\u57df\uff1a<\/p>\n<h3>11. <strong>\u4e8c\u9879\u5f0f\u5b9a\u7406<\/strong><\/h3>\n<p>$$<br \/>\n(a + b)^n = \\sum_{k=0}^{n} \\binom{n}{k} a^{n-k} b^k<br \/>\n$$<br \/>\n\u5176\u4e2d $$ \\binom{n}{k} $$ \u662f\u4e8c\u9879\u5f0f\u7cfb\u6570\uff0c\u8868\u793a $$ n $$ \u4e2a\u5143\u7d20\u4e2d\u53d6 $$ k $$ \u4e2a\u7684\u7ec4\u5408\u6570\u3002<\/p>\n<h3>12. <strong>\u6307\u6570\u51fd\u6570\u7684\u6027\u8d28<\/strong><\/h3>\n<p>$$<br \/>\na^m \\cdot a^n = a^{m+n}<br \/>\n\\quad \\frac{a^m}{a^n} = a^{m-n}<br \/>\n\\quad (a^m)^n = a^{m \\cdot n}<br \/>\n$$<\/p>\n<h3>13. <strong>\u5bf9\u6570\u51fd\u6570\u7684\u6027\u8d28<\/strong><\/h3>\n<p>$$<br \/>\n\\log_a(xy) = \\log_a x + \\log_a y<br \/>\n\\quad \\log_a\\left(\\frac{x}{y}\\right) = \\log_a x - \\log_a y<br \/>\n\\quad \\log_a x^b = b \\log_a x<br \/>\n$$<\/p>\n<h3>14. <strong>\u4e09\u89d2\u51fd\u6570\u6052\u7b49\u5f0f<\/strong><\/h3>\n<ul>\n<li>\n<p><strong>\u548c\u89d2\u516c\u5f0f<\/strong><br \/>\n$$<br \/>\n\\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B<br \/>\n$$<br \/>\n$$<br \/>\n\\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p><strong>\u500d\u89d2\u516c\u5f0f<\/strong><br \/>\n$$<br \/>\n\\sin 2\\theta = 2 \\sin \\theta \\cos \\theta<br \/>\n\\quad \\cos 2\\theta = \\cos^2 \\theta - \\sin^2 \\theta<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p><strong>\u4e09\u89d2\u51fd\u6570\u7684\u5e73\u65b9\u5173\u7cfb<\/strong><br \/>\n$$<br \/>\n\\sin^2 \\theta + \\cos^2 \\theta = 1<br \/>\n$$<\/p>\n<\/li>\n<\/ul>\n<h3>15. <strong>\u5bf9\u79f0\u6027\u5b9a\u7406\uff08\u4f59\u5f26\u5b9a\u7406\uff09<\/strong><\/h3>\n<p>$$<br \/>\nc^2 = a^2 + b^2 - 2ab \\cos C<br \/>\n$$<br \/>\n\u9002\u7528\u4e8e\u4efb\u610f\u4e09\u89d2\u5f62\uff0c( a ), ( b ), ( c ) \u662f\u4e09\u89d2\u5f62\u7684\u4e09\u6761\u8fb9\uff0c( C ) \u662f ( a ) \u548c ( b ) \u6240\u5939\u7684\u89d2\u3002<\/p>\n<h3>16. <strong>\u6b63\u5f26\u5b9a\u7406<\/strong><\/h3>\n<p>$$<br \/>\n\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}<br \/>\n$$<br \/>\n\u9002\u7528\u4e8e\u4efb\u610f\u4e09\u89d2\u5f62\uff0c( A ), ( B ), ( C ) \u5206\u522b\u662f ( a ), ( b ), ( c ) \u6240\u5bf9\u7684\u89d2\u3002<\/p>\n<h3>17. <strong>\u6392\u5217\u6570\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\nP(n, k) = \\frac{n!}{(n-k)!}<br \/>\n$$<br \/>\n\u5176\u4e2d ( P(n, k) ) \u8868\u793a\u4ece ( n ) \u4e2a\u5143\u7d20\u4e2d\u53d6\u51fa ( k ) \u4e2a\u5e76\u6392\u5217\u7684\u65b9\u5f0f\u6570\u3002<\/p>\n<h3>18. <strong>\u7ec4\u5408\u6570\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\nC(n, k) = \\binom{n}{k} = \\frac{n!}{k!(n-k)!}<br \/>\n$$<br \/>\n\u5176\u4e2d ( C(n, k) ) \u8868\u793a\u4ece ( n ) \u4e2a\u5143\u7d20\u4e2d\u53d6\u51fa ( k ) \u4e2a\u5e76\u4e0d\u8003\u8651\u987a\u5e8f\u7684\u7ec4\u5408\u65b9\u5f0f\u6570\u3002<\/p>\n<h3>19. <strong>\u5bfc\u6570\u7684\u5b9a\u4e49<\/strong><\/h3>\n<p>$$<br \/>\nf'(x) = \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x) - f(x)}{\\Delta x}<br \/>\n$$<br \/>\n\u5bfc\u6570\u8868\u793a\u51fd\u6570\u7684\u53d8\u5316\u7387\u3002<\/p>\n<h3>20. <strong>\u79ef\u5206\u7684\u5b9a\u4e49<\/strong><\/h3>\n<p>$$<br \/>\n\\int<em>a^b f(x)\\, dx = \\lim<\/em>{n \\to \\infty} \\sum_{i=1}^n f(x_i^*) \\Delta x<br \/>\n$$<br \/>\n\u79ef\u5206\u8868\u793a\u51fd\u6570\u5728\u533a\u95f4 ([a, b]) \u4e0a\u7684\u9762\u79ef\u3002<\/p>\n<h3>21. <strong>\u5e38\u89c1\u51fd\u6570\u7684\u5bfc\u6570<\/strong><\/h3>\n<ul>\n<li>( \\frac{d}{dx} (x^n) = n x^{n-1} )<\/li>\n<li>( \\frac{d}{dx} (\\sin x) = \\cos x )<\/li>\n<li>( \\frac{d}{dx} (\\cos x) = -\\sin x )<\/li>\n<li>( \\frac{d}{dx} (\\ln x) = \\frac{1}{x} )<\/li>\n<li>( \\frac{d}{dx} (e^x) = e^x )<\/li>\n<\/ul>\n<h3>22. <strong>\u5e38\u89c1\u51fd\u6570\u7684\u79ef\u5206<\/strong><\/h3>\n<ul>\n<li>( \\int x^n\\, dx = \\frac{x^{n+1}}{n+1} + C )<\/li>\n<li>( \\int \\sin x\\, dx = -\\cos x + C )<\/li>\n<li>( \\int \\cos x\\, dx = \\sin x + C )<\/li>\n<li>( \\int \\frac{1}{x}\\, dx = \\ln |x| + C )<\/li>\n<li>( \\int e^x\\, dx = e^x + C )<\/li>\n<\/ul>\n<h3>23. <strong>\u9762\u79ef\u516c\u5f0f<\/strong><\/h3>\n<ul>\n<li>\n<p><strong>\u77e9\u5f62<\/strong>:<br \/>\n$$<br \/>\nA = l \\times w<br \/>\n$$<br \/>\n\u5176\u4e2d ( l ) \u662f\u957f\u5ea6\uff0c( w ) \u662f\u5bbd\u5ea6\u3002<\/p>\n<\/li>\n<li>\n<p><strong>\u4e09\u89d2\u5f62<\/strong>:<br \/>\n$$<br \/>\nA = \\frac{1}{2} \\times \\text{\u5e95} \\times \\text{\u9ad8}<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p><strong>\u68af\u5f62<\/strong>:<br \/>\n$$<br \/>\nA = \\frac{1}{2} \\times (\\text{\u4e0a\u5e95} + \\text{\u4e0b\u5e95}) \\times \\text{\u9ad8}<br \/>\n$$<\/p>\n<\/li>\n<\/ul>\n<h3>24. <strong>\u4f53\u79ef\u516c\u5f0f<\/strong><\/h3>\n<ul>\n<li>\n<p><strong>\u957f\u65b9\u4f53<\/strong>:<br \/>\n$$<br \/>\nV = l \\times w \\times h<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p><strong>\u5706\u67f1\u4f53<\/strong>:<br \/>\n$$<br \/>\nV = \\pi r^2 h<br \/>\n$$<\/p>\n<\/li>\n<li>\n<p><strong>\u7403\u4f53<\/strong>:<br \/>\n$$<br \/>\nV = \\frac{4}{3} \\pi r^3<br \/>\n$$<\/p>\n<\/li>\n<\/ul>\n<h3>25. <strong>\u6b27\u62c9\u516c\u5f0f<\/strong><\/h3>\n<p>$$<br \/>\ne^{i \\theta} = \\cos \\theta + i \\sin \\theta<br \/>\n$$<br \/>\n\u662f\u590d\u6570\u4e0e\u4e09\u89d2\u51fd\u6570\u7684\u8457\u540d\u5173\u7cfb\u5f0f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Markdown\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528 LaTeX \u6765\u5199\u6570\u5b66\u516c\u5f0f\u3002\u53ea\u9700\u5728\u516c\u5f0f\u524d\u540e\u52a0\u4e0a\u7f8e\u5143\u7b26\u53f7 &#8230; \u6765\u8868\u793a\u884c\u5185\u516c\u5f0f\uff0c\u6216\u4f7f\u7528 &#8230; \u6765\u8868\u793a\u72ec\u7acb\u884c\u7684\u516c\u5f0f\u3002 \u53c2\u89c1 https:\/\/jupyterbook.org\/en\/stable\/content\/math.html \u4ee5\u4e0b\u662f\u4e00\u4e9b\u4e2d\u5b66\u6570\u5b66\u5e38\u7528\u7684\u516c\u5f0f\u793a\u4f8b\uff1a 1. \u4e8c\u6b21\u65b9\u7a0b\u6c42\u6839\u516c\u5f0f $$ x = \\frac{-b \\pm \\sqrt{b^2 &#8211; 4ac}}{2a} $$ \u7528\u4e8e\u6c42\u89e3\u4e8c\u6b21\u65b9\u7a0b $$ax^2 + bx + c = 0$$ 2. \u52fe\u80a1\u5b9a\u7406 $$ a^2 + b^2 = c^2 $$ \u5176\u4e2d ( a ) \u548c ( b ) \u662f\u76f4\u89d2\u4e09\u89d2\u5f62\u7684\u4e24\u6761\u76f4\u89d2\u8fb9\uff0c( c ) \u662f\u659c\u8fb9\u3002 3. \u4e58\u6cd5\u516c\u5f0f\uff08\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f\uff09 $$ (a + [&hellip;] <a class=\"read-more\" href=\"https:\/\/www.fanyamin.com\/wordpress\/?p=1420\" title=\"Permanent Link to: \u7528 latex \u6765\u8868\u793a\u6570\u5b66\u516c\u5f0f\">&rarr;Read&nbsp;more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-1420","post","type-post","status-publish","format-standard","hentry","category-5"],"_links":{"self":[{"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1420"}],"collection":[{"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1420"}],"version-history":[{"count":2,"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1420\/revisions"}],"predecessor-version":[{"id":1422,"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1420\/revisions\/1422"}],"wp:attachment":[{"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1420"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1420"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fanyamin.com\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}