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parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.8388800000000001em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"and some boundary conditions, which we will ignore for now - more \", mdx(\"a\", {\n    parentName: \"p\",\n    \"href\": \"/knowledge_base/boundary_conditions\"\n  }, \"here\")), mdx(\"p\", null, \"The above set of equations are called the \", mdx(\"em\", {\n    parentName: \"p\"\n  }, \"strong form\"), \" of the wave equation.\\nFinite-element methods, however solve the \", mdx(\"em\", {\n    parentName: \"p\"\n  }, \"weak\"), \" or \", mdx(\"em\", {\n    parentName: \"p\"\n  }, \"variational form\"), \" of the PDE.\"), mdx(\"h2\", null, \"Weak form\"), mdx(\"p\", null, \"The weak for of the wave equation is obtained by multiplying the equation with a so-called \", mdx(\"em\", {\n    parentName: \"p\"\n  }, \"test function\"), \" \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C6\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\varphi(\\\\mathbf{x})\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C6\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"))))), \" and integrating in space:\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msub\"\n  }, \"\\u222B\"), mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u03A9\")), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"c\"), mdx(\"mrow\", {\n    parentName: \"msup\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"1\"))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msubsup\"\n  }, \"t\"), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C6\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msub\"\n  }, \"\\u222B\"), mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u03A9\")), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u0394\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C6\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msub\"\n  }, \"\\u222B\"), mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u03A9\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"f\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C6\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \".\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"  \\\\int_\\\\Omega c^{-1}(\\\\mathbf{x}) \\\\partial_t^2 u(\\\\mathbf{x},t)\\\\, \\\\varphi(\\\\mathbf{x})\\\\,d\\\\mathbf{x} - \\\\int_\\\\Omega \\\\Delta u(\\\\mathbf{x},t)\\\\,\\\\varphi(\\\\mathbf{x})\\\\,d\\\\mathbf{x}\\n  =\\\\int_\\\\Omega f(\\\\mathbf{x},t)\\\\,\\\\varphi(\\\\mathbf{x})\\\\,d\\\\mathbf{x}.\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.27195em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"-0.433619em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u03A9\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"c\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.864108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"1\"))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C6\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.27195em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: 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parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, 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  \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"-0.425842em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u03A9\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2207\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": 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\"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C6\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"where the last term is a surface integral along the boundary \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u03A9\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\partial\\\\Omega\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.69444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u03A9\"))))), \" of the domain \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u03A9\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\Omega\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: 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\"\\u20D7\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\vec{\\\\mathbf{n}}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.72744em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.72744em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"n\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.01344em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.2355em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))))))), \" is the unit normal point outward of the domain. 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parentName: \"msup\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"1\"))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msubsup\"\n  }, \"t\"), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: 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parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.27195em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"-0.425842em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u03A9\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2207\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u22C5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord accent\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.72744em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"n\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.01344em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"accent-body\",\n    \"style\": {\n      \"left\": \"-0.2355em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"overlay\",\n    \"style\": {\n      \"height\": \"0.714em\",\n      \"width\": \"0.471em\"\n    }\n  }, mdx(\"svg\", {\n    parentName: \"span\",\n    \"width\": \"0.471em\",\n    \"height\": \"0.714em\",\n    \"style\": {\n      \"width\": \"0.471em\"\n    },\n    \"viewBox\": \"0 0 471 714\",\n    \"preserveAspectRatio\": \"xMinYMin\"\n  }, mdx(\"path\", {\n    parentName: \"svg\",\n    \"d\": \"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\\nc-16-25.333-24-45-24-59z\"\n  }))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C6\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.27195em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"-0.433619em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u03A9\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10764em\"\n    }\n  }, \"f\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C6\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"x\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"A function \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"))))), \" that satisfies this weak form for any choice of the test function \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C6\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\varphi\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.625em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C6\"))))), \" at all times \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.61508em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"))))), \" is called a \", mdx(\"em\", {\n    parentName: \"p\"\n  }, \"weak solution\"), \". Under certain mathematical conditions, \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"))))), \" also satisfies the strong form above. We spare the details here, and just assume that both solutions are the same.\"));\n}\n;\nMDXContent.isMDXComponent = true;"},"site":{"siteMetadata":{"salvusDocVersions":{"current":"2026.5.0"}}}},"pageContext":{"id":"9d62be38-ea4b-5e5b-b113-cd691126cc23"}},
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