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2207--nh-l-green-la-gi [2018/11/07 17:11] (current)
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 +<​HTML><​br><​div id="​mw-content-text"​ lang="​vi"​ dir="​ltr"><​div class="​mw-parser-output"><​p>​Trong toán học, <​b>​định lý Green'</​b>​ đưa ra mối liên hệ giữa tích phân đường quanh một đường cong khép kín  <​i>​C</​i>​ vàa tích phân mặt trên một miền <​i>​D</​i> ​ bao quanh bởi <​i>​C</​i>​. Đây là trường hợp đặc biệt trong không gian 2 chiều của  định lý Stokes, và được đặt tên theo nhà toán học người Anh tên George Green.
 +</p>
  
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 +<​p><​i>​C</​i>​ là một đường đơn đóng có định hướng dương trong mặt phẳng <​b><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbb {R} }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​double-struck">​R</​mi></​mrow></​mstyle></​mrow>​{displaystyle mathbb {R} }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​786849c765da7a84dbc3cce43e96aad58a5868dc"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​1.678ex;​ height:​2.176ex;"​ alt="​{displaystyle mathbb {R} }"/></​span><​sup>​2</​sup></​b>,​ và <​i>​D</​i>​ là miền được bao quanh bởi <​i>​C</​i>​. Nếu <​i>​L</​i>​ và <​i>​M</​i>​ là các hàm số với biến (<​i>​x</​i>,​ <​i>​y</​i>​) được định nghĩa trên <a href="​http://​vi.wikipedia.org/​w/​index.php?​title=M%E1%BB%9F&​amp;​action=edit&​amp;​redlink=1"​ class="​new"​ title="​Mở (trang chưa được viết)">​miền mở chứa <​i>​D</​i>​ và có các đạo hàm riêng phần liên tục trên đó, thì<sup id="​cite_ref-1"​ class="​reference">​[1]</​sup><​sup id="​cite_ref-2"​ class="​reference">​[2]</​sup></​p>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle oint _{C}(P,​mathrm {d} x+Q,mathrm {d} y)=iint _{D}left({frac {partial Q}{partial x}}-{frac {partial P}{partial y}}right),​mathrm {d} x,mathrm {d} y.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​P</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​x</​mi><​mo>​+</​mo><​mi>​Q</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​D</​mi></​mrow></​msub><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​Q</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​P</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​y</​mi></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​y</​mi><​mo>​.</​mo></​mstyle></​mrow>​{displaystyle oint _{C}(P,​mathrm {d} x+Q,mathrm {d} y)=iint _{D}left({frac {partial Q}{partial x}}-{frac {partial P}{partial y}}right),​mathrm {d} x,mathrm {d} y.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​c1df09c7a73ba0aa6fcc293432af40488165373e"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​47.898ex;​ height:​6.176ex;"​ alt="​{displaystyle oint _{C}(P,​mathrm {d} x+Q,mathrm {d} y)=iint _{D}left({frac {partial Q}{partial x}}-{frac {partial P}{partial y}}right),​mathrm {d} x,mathrm {d} y.}"/></​span></​dd></​dl><​h2><​span id="​M.E1.BB.91i_li.C3.AAn_h.E1.BB.87_v.E1.BB.9Bi_.C4.91.E1.BB.8Bnh_l.C3.BD_Stokes"/><​span class="​mw-headline"​ id="​Mối_liên_hệ_với_định_lý_Stokes">​Mối liên hệ với định lý Stokes</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span><​a href="​http://​vi.wikipedia.org/​w/​index.php?​title=%C4%90%E1%BB%8Bnh_l%C3%BD_Green&​amp;​veaction=edit&​amp;​section=2"​ class="​mw-editsection-visualeditor"​ title="​Sửa đổi phần “Mối liên hệ với định lý Stokes”">​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h2>​
 +<​p>​Định lý Green là một trường hợp đặc biệt của định lý Stokes, khi áp dụng trên mặt phẳng-<​i>​xy</​i>:​
 +</​p><​p>​Chúng ta có thể mở rộng trường 2 chiều thành một trường trong không gian 3 chiều với thành phần <​i>​z</​i>​ luôn bằng 0. Gọi <​b>​F</​b>​ là hàm số vector định nghĩa bởi <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbf {F} =(L,​M,​0)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​=</​mo><​mo stretchy="​false">​(</​mo><​mi>​L</​mi><​mo>,</​mo><​mi>​M</​mi><​mo>,</​mo><​mn>​0</​mn><​mo stretchy="​false">​)</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbf {F} =(L,​M,​0)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​0af6fb0321869c4df3e20264886eadca117911bc"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​13.846ex;​ height:​2.843ex;"​ alt="​{displaystyle mathbf {F} =(L,​M,​0)}"/></​span>​. ​ Bắt đầu với vế trái của định lý Green:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle oint _{C}(P,​dx+Q,​dy)=oint _{C}(L,​M,​0)cdot (dx,​dy,​dz)=oint _{C}mathbf {F} cdot dmathbf {r} .}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​P</​mi><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​x</​mi><​mo>​+</​mo><​mi>​Q</​mi><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​L</​mi><​mo>,</​mo><​mi>​M</​mi><​mo>,</​mo><​mn>​0</​mn><​mo stretchy="​false">​)</​mo><​mo>​⋅<​!-- &sdot; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​d</​mi><​mi>​x</​mi><​mo>,</​mo><​mi>​d</​mi><​mi>​y</​mi><​mo>,</​mo><​mi>​d</​mi><​mi>​z</​mi><​mo stretchy="​false">​)</​mo><​mo>​=</​mo><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mi>​d</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​r</​mi></​mrow><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle oint _{C}(P,​dx+Q,​dy)=oint _{C}(L,​M,​0)cdot (dx,​dy,​dz)=oint _{C}mathbf {F} cdot dmathbf {r} .}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​d880fa01471225344bc1aa218329a0de43d5abba"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​64.249ex;​ height:​6.009ex;"​ alt="​{displaystyle oint _{C}(P,​dx+Q,​dy)=oint _{C}(L,​M,​0)cdot (dx,​dy,​dz)=oint _{C}mathbf {F} cdot dmathbf {r} .}"/></​span></​dd></​dl><​p>​Theo định lý Stokes thì:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle oint _{C}mathbf {F} cdot dmathbf {r} =iint _{S}nabla times mathbf {F} cdot mathbf {hat {n}} ,​dS.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mi>​d</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​r</​mi></​mrow><​mo>​=</​mo><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​S</​mi></​mrow></​msub><​mi mathvariant="​normal">​∇<​!-- &nabla; --></​mi><​mo>​×<​!-- &times; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​S</​mi><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle oint _{C}mathbf {F} cdot dmathbf {r} =iint _{S}nabla times mathbf {F} cdot mathbf {hat {n}} ,​dS.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​1634b3676e22e0b0ad5f53b218fba4cecb909243"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​31.927ex;​ height:​6.009ex;"​ alt="​{displaystyle oint _{C}mathbf {F} cdot dmathbf {r} =iint _{S}nabla times mathbf {F} cdot mathbf {hat {n}} ,​dS.}"/></​span></​dd></​dl><​p>​Mặt <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle S}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​S</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle S}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4611d85173cd3b508e67077d4a1252c9c05abca2"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​1.499ex;​ height:​2.176ex;"​ alt="​S"/></​span>​ chỉ là một miền <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle D}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​D</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle D}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​f34a0c600395e5d4345287e21fb26efd386990e6"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​1.924ex;​ height:​2.176ex;"​ alt="​D"/></​span>​ trong mặt phẳng, với vector định chuẩn <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbf {hat {n}} }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbf {hat {n}} }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4eb84e133d15551d660800ec29b44783ff36e19d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​1.485ex;​ height:​2.343ex;"​ alt="​{displaystyle mathbf {hat {n}} }"/></​span>​ hướng lên (theo hướng <​i>​z</​i>​) để trùng với "​định hướng dương"​ trong cả hai định lý.
 +</​p><​p>​Biểu thức bên trong tích phân trở thành ​
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle nabla times mathbf {F} cdot mathbf {hat {n}} =left[left({frac {partial 0}{partial y}}-{frac {partial M}{partial z}}right)mathbf {i} +left({frac {partial L}{partial z}}-{frac {partial 0}{partial x}}right)mathbf {j} +left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right)mathbf {k} right]cdot mathbf {k} =left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right).}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi mathvariant="​normal">​∇<​!-- &nabla; --></​mi><​mo>​×<​!-- &times; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow><​mo>​=</​mo><​mrow><​mo>​[</​mo><​mrow><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mn>​0</​mn></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​y</​mi></​mrow></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​M</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​z</​mi></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​i</​mi></​mrow><​mo>​+</​mo><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​L</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​z</​mi></​mrow></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mn>​0</​mn></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​j</​mi></​mrow><​mo>​+</​mo><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​M</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​L</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​y</​mi></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​k</​mi></​mrow></​mrow><​mo>​]</​mo></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​k</​mi></​mrow><​mo>​=</​mo><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​M</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​L</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​y</​mi></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle nabla times mathbf {F} cdot mathbf {hat {n}} =left[left({frac {partial 0}{partial y}}-{frac {partial M}{partial z}}right)mathbf {i} +left({frac {partial L}{partial z}}-{frac {partial 0}{partial x}}right)mathbf {j} +left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right)mathbf {k} right]cdot mathbf {k} =left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right).}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​77a90d83bc979c291f726a35cf8811cbf7706540"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​89.058ex;​ height:​6.176ex;"​ alt="​{displaystyle nabla times mathbf {F} cdot mathbf {hat {n}} =left[left({frac {partial 0}{partial y}}-{frac {partial M}{partial z}}right)mathbf {i} +left({frac {partial L}{partial z}}-{frac {partial 0}{partial x}}right)mathbf {j} +left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right)mathbf {k} right]cdot mathbf {k} =left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right).}"/></​span></​dd></​dl><​p>​Do đó mà ta sẽ được vế phải của định lý Green
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle iint _{S}nabla times mathbf {F} cdot mathbf {hat {n}} ,dS=iint _{D}left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right),​dA.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​S</​mi></​mrow></​msub><​mi mathvariant="​normal">​∇<​!-- &nabla; --></​mi><​mo>​×<​!-- &times; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​S</​mi><​mo>​=</​mo><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​D</​mi></​mrow></​msub><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​M</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​L</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​y</​mi></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​A</​mi><​mo>​.</​mo></​mstyle></​mrow>​{displaystyle iint _{S}nabla times mathbf {F} cdot mathbf {hat {n}} ,dS=iint _{D}left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right),​dA.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​e157f5541999e99d36425724121d3b687a7d3c81"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​43.496ex;​ height:​6.176ex;"​ alt="​{displaystyle iint _{S}nabla times mathbf {F} cdot mathbf {hat {n}} ,dS=iint _{D}left({frac {partial M}{partial x}}-{frac {partial L}{partial y}}right),​dA.}"/></​span></​dd></​dl><​h2><​span id="​M.E1.BB.91i_li.C3.AAn_quan_v.E1.BB.9Bi_.C4.91.E1.BB.8Bnh_l.C3.BD_Gauss"/><​span class="​mw-headline"​ id="​Mối_liên_quan_với_định_lý_Gauss">​Mối liên quan với định lý Gauss</​span><​span class="​mw-editsection"><​span class="​mw-editsection-bracket">​[</​span><​a href="​http://​vi.wikipedia.org/​w/​index.php?​title=%C4%90%E1%BB%8Bnh_l%C3%BD_Green&​amp;​veaction=edit&​amp;​section=3"​ class="​mw-editsection-visualeditor"​ title="​Sửa đổi phần “Mối liên quan với định lý Gauss”">​sửa<​span class="​mw-editsection-divider">​ | </​span>​sửa mã nguồn<​span class="​mw-editsection-bracket">​]</​span></​span></​h2>​
 +<​p>​Nếu chỉ xét các trường vectơ trong không gian 2 chiều,
 +định lý Green là tương đương với phiên bản 2 chiều sau đây của định lý Gauss:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle iint _{D}left(nabla cdot mathbf {F} right)dA=oint _{C}mathbf {F} cdot mathbf {hat {n}} ,​ds,​}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​D</​mi></​mrow></​msub><​mrow><​mo>​(</​mo><​mrow><​mi mathvariant="​normal">​∇<​!-- &nabla; --></​mi><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow></​mrow><​mo>​)</​mo></​mrow><​mi>​d</​mi><​mi>​A</​mi><​mo>​=</​mo><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​s</​mi><​mo>,</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle iint _{D}left(nabla cdot mathbf {F} right)dA=oint _{C}mathbf {F} cdot mathbf {hat {n}} ,​ds,​}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​506548f476d5c43024f657cbc091823fe497877c"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​31.815ex;​ height:​6.009ex;"​ alt="​{displaystyle iint _{D}left(nabla cdot mathbf {F} right)dA=oint _{C}mathbf {F} cdot mathbf {hat {n}} ,​ds,​}"/></​span></​dd></​dl><​p>​với <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbf {hat {n}} }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbf {hat {n}} }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4eb84e133d15551d660800ec29b44783ff36e19d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​1.485ex;​ height:​2.343ex;"​ alt="​{displaystyle mathbf {hat {n}} }"/></​span>​ là véc tơ định chuẩn hướng ra ngoài trên biên.
 +</​p><​p>​Để thấy điều này, xét vec tơ định chuẩn <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbf {hat {n}} }"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbf {hat {n}} }</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​4eb84e133d15551d660800ec29b44783ff36e19d"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.338ex; width:​1.485ex;​ height:​2.343ex;"​ alt="​{displaystyle mathbf {hat {n}} }"/></​span>​ ở tay phải của phương trình. Bởi vì trong định lý Green <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle dmathbf {r} =(dx,​dy)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​d</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​r</​mi></​mrow><​mo>​=</​mo><​mo stretchy="​false">​(</​mo><​mi>​d</​mi><​mi>​x</​mi><​mo>,</​mo><​mi>​d</​mi><​mi>​y</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle dmathbf {r} =(dx,​dy)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​f8ec46e15f5fa115c303a08f5c74e12bb9cfdda9"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​13.177ex;​ height:​2.843ex;"​ alt="​{displaystyle dmathbf {r} =(dx,​dy)}"/></​span>​ là một vecto đi theo hướng tiếp tuyến với đường cong, và đường cong C được định hướng dương (ngược chiều kim đồng hồ) dọc theo biên, vectơ định chuẩn hướng ra ngoài sẽ chỉ vuông góc  90° về phía phải, và sẽ là <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle (dy,​-dx)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mo stretchy="​false">​(</​mo><​mi>​d</​mi><​mi>​y</​mi><​mo>,</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​d</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle (dy,​-dx)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​e6b38e701840ab8cc347a7a136eeae447f7e0792"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​9.568ex;​ height:​2.843ex;"​ alt="​{displaystyle (dy,​-dx)}"/></​span>​. Chiều dài của vec tơ này là <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {sqrt {dx^{2}+dy^{2}}}=ds}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​msqrt><​mi>​d</​mi><​msup><​mi>​x</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup><​mo>​+</​mo><​mi>​d</​mi><​msup><​mi>​y</​mi><​mrow class="​MJX-TeXAtom-ORD"><​mn>​2</​mn></​mrow></​msup></​msqrt></​mrow><​mo>​=</​mo><​mi>​d</​mi><​mi>​s</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {sqrt {dx^{2}+dy^{2}}}=ds}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9a55b66b50280d8988e946cd5ac43ef812c0d1e2"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -1.671ex; width:​17.599ex;​ height:​4.843ex;"​ alt="​{displaystyle {sqrt {dx^{2}+dy^{2}}}=ds}"/></​span>​. Do vậy <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbf {hat {n}} ,​ds=(dy,​-dx).}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​s</​mi><​mo>​=</​mo><​mo stretchy="​false">​(</​mo><​mi>​d</​mi><​mi>​y</​mi><​mo>,</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​d</​mi><​mi>​x</​mi><​mo stretchy="​false">​)</​mo><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbf {hat {n}} ,​ds=(dy,​-dx).}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​8e7f107f803a549fbd4df9c58f2a5eb89ac93039"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​17.492ex;​ height:​2.843ex;"​ alt="​{displaystyle mathbf {hat {n}} ,​ds=(dy,​-dx).}"/></​span>​
 +</​p><​p>​Bây giờ hãy để <span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle mathbf {F} =(P,​Q)}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​=</​mo><​mo stretchy="​false">​(</​mo><​mi>​P</​mi><​mo>,</​mo><​mi>​Q</​mi><​mo stretchy="​false">​)</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle mathbf {F} =(P,​Q)}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​7e4c21c5ce67e62dce3dd37ce5411d1ec82efc44"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -0.838ex; width:​11.208ex;​ height:​2.843ex;"​ alt="​{displaystyle mathbf {F} =(P,​Q)}"/></​span>​. Khi đó vế phải sẽ trở thành
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle oint _{C}mathbf {F} cdot mathbf {hat {n}} ,ds=oint _{C}Pdy-Qdx}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mrow class="​MJX-TeXAtom-ORD"><​mover><​mi mathvariant="​bold">​n</​mi><​mo mathvariant="​bold"​ stretchy="​false">​^<​!-- ^ --></​mo></​mover></​mrow></​mrow><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​s</​mi><​mo>​=</​mo><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​P</​mi><​mi>​d</​mi><​mi>​y</​mi><​mo>​−<​!-- &minus; --></​mo><​mi>​Q</​mi><​mi>​d</​mi><​mi>​x</​mi></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle oint _{C}mathbf {F} cdot mathbf {hat {n}} ,ds=oint _{C}Pdy-Qdx}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​9d26db2b8ad4f2952dcb0faa31f98c37a771342a"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​33.135ex;​ height:​6.009ex;"​ alt="​{displaystyle oint _{C}mathbf {F} cdot mathbf {hat {n}} ,ds=oint _{C}Pdy-Qdx}"/></​span></​dd></​dl><​p>​mà do định lý Green sẽ trở thành
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle oint _{C}-Qdx+Pdy=iint _{D}left({frac {partial P}{partial x}}+{frac {partial Q}{partial y}}right),​dA=iint _{D}left(nabla cdot mathbf {F} right)dA.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​−<​!-- &minus; --></​mo><​mi>​Q</​mi><​mi>​d</​mi><​mi>​x</​mi><​mo>​+</​mo><​mi>​P</​mi><​mi>​d</​mi><​mi>​y</​mi><​mo>​=</​mo><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​D</​mi></​mrow></​msub><​mrow><​mo>​(</​mo><​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​P</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​+</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​Q</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​y</​mi></​mrow></​mfrac></​mrow></​mrow><​mo>​)</​mo></​mrow><​mspace width="​thinmathspace"/><​mi>​d</​mi><​mi>​A</​mi><​mo>​=</​mo><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​D</​mi></​mrow></​msub><​mrow><​mo>​(</​mo><​mrow><​mi mathvariant="​normal">​∇<​!-- &nabla; --></​mi><​mo>​⋅<​!-- &sdot; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​bold">​F</​mi></​mrow></​mrow><​mo>​)</​mo></​mrow><​mi>​d</​mi><​mi>​A</​mi><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle oint _{C}-Qdx+Pdy=iint _{D}left({frac {partial P}{partial x}}+{frac {partial Q}{partial y}}right),​dA=iint _{D}left(nabla cdot mathbf {F} right)dA.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​e6b2703cf22755f2486065189a10164bf71ad07b"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​63.554ex;​ height:​6.176ex;"​ alt="​{displaystyle oint _{C}-Qdx+Pdy=iint _{D}left({frac {partial P}{partial x}}+{frac {partial Q}{partial y}}right),​dA=iint _{D}left(nabla cdot mathbf {F} right)dA.}"/></​span></​dd></​dl><​p>​Điều ngược lại cũng có thể được chứng minh một cách dễ dàng.
 +</p>
 +
 +<​p>​Định lý Green có thể được sử dụng để tính diện tích sử dụng tích phân đường.<​sup id="​cite_ref-stuart_3-0"​ class="​reference">​[3]</​sup>​ Diện tích của miền D được cho bởi:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle A=iint _{D}mathrm {d} A.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​A</​mi><​mo>​=</​mo><​msub><​mo>​∬<​!-- &#8748; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mi>​D</​mi></​mrow></​msub><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​A</​mi><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle A=iint _{D}mathrm {d} A.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​635c60ff65f2723b523fbcd6e53091a1a2dfded0"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.338ex; width:​13.023ex;​ height:​5.676ex;"​ alt="​{displaystyle A=iint _{D}mathrm {d} A.}"/></​span></​dd></​dl><​p>​Miễn là chúng ta chọn được L và M sao cho:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle {frac {partial M}{partial x}}-{frac {partial L}{partial y}}=1.}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​M</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​x</​mi></​mrow></​mfrac></​mrow><​mo>​−<​!-- &minus; --></​mo><​mrow class="​MJX-TeXAtom-ORD"><​mfrac><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​L</​mi></​mrow><​mrow><​mi mathvariant="​normal">​∂<​!-- &part; --></​mi><​mi>​y</​mi></​mrow></​mfrac></​mrow><​mo>​=</​mo><​mn>​1.</​mn></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle {frac {partial M}{partial x}}-{frac {partial L}{partial y}}=1.}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​a5d8de05632865f4280bd0d851ad224a1d92c5da"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​16.082ex;​ height:​6.009ex;"​ alt="​{displaystyle {frac {partial M}{partial x}}-{frac {partial L}{partial y}}=1.}"/></​span></​dd></​dl><​p>​Diện tích sẽ được cho bởi công thức sau:
 +</p>
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle A=oint _{C}(L,​mathrm {d} x+M,mathrm {d} y).}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​A</​mi><​mo>​=</​mo><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mi>​L</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​x</​mi><​mo>​+</​mo><​mi>​M</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle A=oint _{C}(L,​mathrm {d} x+M,mathrm {d} y).}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​c78b3b17a3d9076222be2ae72fe7ede2a651b972"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​25.198ex;​ height:​6.009ex;"​ alt="​{displaystyle A=oint _{C}(L,​mathrm {d} x+M,mathrm {d} y).}"/></​span></​dd></​dl><​p>​Các công thức cho diện tích của D bao gồm:<​sup id="​cite_ref-stuart_3-1"​ class="​reference">​[3]</​sup></​p>​
 +<​dl><​dd><​span class="​mwe-math-element"><​span class="​mwe-math-mathml-inline mwe-math-mathml-a11y"​ style="​display:​ none;"><​math xmlns="​http://​www.w3.org/​1998/​Math/​MathML"​ alttext="​{displaystyle A=oint _{C}x,​mathrm {d} y=-oint _{C}y,​mathrm {d} x={tfrac {1}{2}}oint _{C}(-y,​mathrm {d} x+x,mathrm {d} y).}"><​semantics><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​true"​ scriptlevel="​0"><​mi>​A</​mi><​mo>​=</​mo><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​y</​mi><​mo>​=</​mo><​mo>​−<​!-- &minus; --></​mo><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mi>​y</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​x</​mi><​mo>​=</​mo><​mrow class="​MJX-TeXAtom-ORD"><​mstyle displaystyle="​false"​ scriptlevel="​0"><​mfrac><​mn>​1</​mn><​mn>​2</​mn></​mfrac></​mstyle></​mrow><​msub><​mrow class="​MJX-TeXAtom-OP MJX-fixedlimits"><​mrow class="​MJX-TeXAtom-VCENTER"><​mstyle mathsize="​2.07em"><​mtext>​∮<​!-- &#8750; --></​mtext><​mspace width="​thinmathspace"/></​mstyle></​mrow><​mspace width="​thinmathspace"/></​mrow><​mrow class="​MJX-TeXAtom-ORD"><​mi>​C</​mi></​mrow></​msub><​mo>​⁡<​!-- &#8289; --></​mo><​mo stretchy="​false">​(</​mo><​mo>​−<​!-- &minus; --></​mo><​mi>​y</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​x</​mi><​mo>​+</​mo><​mi>​x</​mi><​mspace width="​thinmathspace"/><​mrow class="​MJX-TeXAtom-ORD"><​mi mathvariant="​normal">​d</​mi></​mrow><​mi>​y</​mi><​mo stretchy="​false">​)</​mo><​mo>​.</​mo></​mstyle></​mrow><​annotation encoding="​application/​x-tex">​{displaystyle A=oint _{C}x,​mathrm {d} y=-oint _{C}y,​mathrm {d} x={tfrac {1}{2}}oint _{C}(-y,​mathrm {d} x+x,mathrm {d} y).}</​annotation></​semantics></​math></​span><​img src="​https://​wikimedia.org/​api/​rest_v1/​media/​math/​render/​svg/​87af5e3108b27cda920591533c224fafad670b8b"​ class="​mwe-math-fallback-image-inline"​ aria-hidden="​true"​ style="​vertical-align:​ -2.505ex; width:​55.389ex;​ height:​6.009ex;"​ alt="​{displaystyle A=oint _{C}x,​mathrm {d} y=-oint _{C}y,​mathrm {d} x={tfrac {1}{2}}oint _{C}(-y,​mathrm {d} x+x,mathrm {d} y).}"/></​span></​dd></​dl>​
 +
 +<div class="​reflist"​ style="​list-style-type:​ decimal;">​
 +<ol class="​references"><​li id="​cite_note-1"><​b>​^</​b>​ <span class="​reference-text">​Mathematical methods for physics and engineering,​ K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3</​span>​
 +</li>
 +<li id="​cite_note-2"><​b>​^</​b>​ <span class="​reference-text">​Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipcshutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7</​span>​
 +</li>
 +<li id="​cite_note-stuart-3">​^ <​sup><​i><​b>​a</​b></​i></​sup>​ <​sup><​i><​b>​ă</​b></​i></​sup>​ <span class="​reference-text"><​span class="​citation book">​Stewart,​ James. <​i>​Calculus</​i>​ (ấn bản 6). Thomson, Brooks/​Cole.</​span><​span title="​ctx_ver=Z39.88-2004&​amp;​rfr_id=info%3Asid%2Fvi.wikipedia.org%3A%C4%90%E1%BB%8Bnh+l%C3%BD+Green&​amp;​rft.au=Stewart%2C+James&​amp;​rft.aufirst=James&​amp;​rft.aulast=Stewart&​amp;​rft.btitle=Calculus&​amp;​rft.edition=6th&​amp;​rft.genre=book&​amp;​rft.pub=Thomson%2C+Brooks%2FCole&​amp;​rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook"​ class="​Z3988"><​span style="​display:​none;">​ </​span></​span></​span>​
 +</li>
 +</​ol></​div>​
 +
 +<​ul><​li><​i>​Calculus (5th edition)</​i>,​ F. Ayres, E. Mendelson, Schuam'​s Outline Series, 2009, ISBN 978-0-07-150861-2.</​li>​
 +<​li><​i>​Advanced Calculus (3rd edition)</​i>,​ R. Wrede, M.R. Spiegel, Schuam'​s Outline Series, 2010, ISBN 978-0-07-162366-7.</​li></​ul>​
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2207--nh-l-green-la-gi.txt · Last modified: 2018/11/07 17:11 (external edit)