課題二十五重積分的應(yīng)用(與“曲面”相關(guān)共71張)

重點：曲面的面積，格林公式，由定積分的元素法引出。

難點：正確求解曲線或曲面積分，由實例講解方法?？倳r數(shù)：4學(xué)時.1、知道第一和第二類曲線積分，第一和第二類曲面積分；2、會求曲線和曲面積分，會用格林公式計算二重積分和高斯公式計算三重積分?！臼谡n時數(shù)】【學(xué)習(xí)目標(biāo)】【重、難點】把定積分的元素法推廣到二重積分的應(yīng)用中.若要計算的某個量U對于閉區(qū)域D具有可加性(即當(dāng)閉區(qū)域D分成許多小閉區(qū)域時，所求量U相應(yīng)地分成許多部分量，且U等于部分量之和)，并且在閉區(qū)域D內(nèi)任取一個直徑很小的閉區(qū)域時，相應(yīng)地部分量可近似地表示為的形式，其中在內(nèi)．這個稱為所求量U的元素，記為，所求量的積分表達(dá)式為一、曲面面積衛(wèi)星1.設(shè)曲面的方程為：如圖，上式是曲面S的面積元素曲面面積公式為：3.設(shè)曲面的方程為：曲面面積公式為：2.設(shè)曲面的方程為：曲面面積公式為：同理可得解解解方程組得兩曲面的交線為圓周在平面上的投影域為（一）P108習(xí)題6.要求光滑曲面的質(zhì)量，下面介紹第一類型曲面積分.在平面上的投影域為所謂曲面光滑即曲面上各點處都有切平面，且當(dāng)點在曲面上連續(xù)移動時，切平面也連續(xù)轉(zhuǎn)動.實例:曲線形構(gòu)件的質(zhì)量使用Guass公式時應(yīng)注意:對坐標(biāo)的曲線積分的計算表達(dá)了空間閉區(qū)域上的三重積分與其邊界曲面上的曲面積分之間的關(guān)系.2．會用格林公式求二重積分和高斯公式求三重積分。實例:流向曲面一側(cè)的流量.表達(dá)了空間閉區(qū)域上的三重積分與其邊界曲面上的曲面積分之間的關(guān)系.要求光滑曲面的質(zhì)量，下面介紹第一類型曲面積分.利用的參數(shù)方程實例:曲線形構(gòu)件的質(zhì)量難點：正確求解曲線或曲面積分，由實例講解方法。結(jié)論：被積函數(shù)相同，起點和終點也相同，但路徑不同而積分結(jié)果相同.對弧長曲線積分的概念和計算實例:曲線形構(gòu)件的質(zhì)量勻質(zhì)曲線形構(gòu)件的質(zhì)量(1)分割(2)求和(3)取極限近似值精確值非勻質(zhì)曲線形構(gòu)件的質(zhì)量求法二、對弧長的曲線積分1.定義被積函數(shù)積分弧段積分和式曲線形構(gòu)件的質(zhì)量2.存在條件：注意：3.對弧長曲線積分的計算定理[例3]解如圖，4.幾何與物理意義[例4]它的對稱軸的轉(zhuǎn)動慣量(設(shè)線密度解取坐標(biāo)系如圖，利用的參數(shù)方程于是計算半徑為中心角為的圓弧對于則實例:

變力沿曲線所作的功常力沿直線所作的功(1)分割變力沿曲線所作的功的算法(2)求和(3)取極限近似值精確值1.定義三、對坐標(biāo)的曲線積分類似地定義2.存在條件第二類曲線積分存在．),(),,(上連續(xù)時，在光滑曲線弧當(dāng)LyxQyxP3.性質(zhì)定理4.對坐標(biāo)的曲線積分的計算特殊情形[例5]解[例6]解結(jié)論：被積函數(shù)相同，起點和終點也相同，但路徑不同而積分結(jié)果相同.1.曲面面積的計算2.對弧長曲線積分的概念和計算3.對坐標(biāo)曲線積分的概念和計算小結(jié)練習(xí)題弧長坐標(biāo)定理5.格林公式邊界曲線L的正向:當(dāng)觀察者沿邊界行走時,區(qū)域D總在他的左邊.xyoLAB解

xyo解

解o曲線由函數(shù)所謂曲面光滑即曲面上各點處都有切平面，且當(dāng)點在曲面上連續(xù)移動時，切平面也連續(xù)轉(zhuǎn)動.要求光滑曲面的質(zhì)量，下面介紹第一類型曲面積分.1.定義四、對面積的曲面積分2.性質(zhì)則按照曲面的不同情況分為以下三種：3.計算法則則解解依對稱性知：觀察以下曲面的側(cè)(假設(shè)曲面是光滑的)曲面分上側(cè)和下側(cè)曲面分內(nèi)側(cè)和外側(cè)曲面?zhèn)鹊拇_定曲面法向量的指向決定曲面的側(cè).決定了側(cè)的曲面稱為有向曲面.實例:流向曲面一側(cè)的流量.流量（2）設(shè)穩(wěn)定流動的不可壓縮流體(假定密度為1)給出,的速度場由有向曲面,函數(shù)都在內(nèi)流向Σ指定側(cè)的流體Σ上連續(xù),求在單位時間.的質(zhì)量Σ是速度場中的一片1.分割為.法向量為.2.求和3.取極限五、對坐標(biāo)的曲面積分1.定義被積函數(shù)積分曲面類似可定義2.存在條件3.一般形式4.物理意義5.性質(zhì)6.計算法注意:對坐標(biāo)的曲面積分,必須注意曲面所取的側(cè).解7.高斯公式Gauss公式的實質(zhì)表達(dá)了空間閉區(qū)域上的三重積分與其邊界曲面上的曲面積分之間的關(guān)系.解利用柱面坐標(biāo)得使用Guass公式時應(yīng)注意:1.格林公式及應(yīng)用2.對面積的曲面積分的概念和計算3.對坐標(biāo)的曲面積分的概念的計算小結(jié)4.高斯公式及應(yīng)用練習(xí)題0通過本課題學(xué)習(xí)，學(xué)生應(yīng)該達(dá)到：1．會求曲線和曲面積分；2．會用格林公式求二重積分和高斯公式求三重積分。（一）P108習(xí)題6.3(1)；（二）P108習(xí)題6.3(2).【授課小結(jié)】【課后練習(xí)】v)z0C3F7IaMdPgSkVnYq$t*x-A1D5G8JbNeQiTlWo#r%u(y+B3E6H9LcOfRjUmYp!s&w)z0C4F7IaMdPhSkVnZq$t*x-A2D5G8KbNeQiTlXo#r%v(y+B3E6I9LcOgRjUmYp!t&w)z1C4F7JaMePhSkWnZq$u*x+A2D5H8KbNfQiUlXo#s%v(y0B3E6I9LdOgRjVmYp!t&w-z1C4G7JaMePhTkWnZr$u*x+A2E5H8KcNfQiUlXp#s%v)y0B3F6IaLdOgSjVmYq!t*w-z1D4G7JbMePhTkWoZr$u(x+A2E5H9KcNfRiUlXp#s&v)y0C3F6IaLdPgSjVnYq!t*w-A1D4G8JbMeQhTlWoZr%u(x+B2E6H9KcOfRiUmXp#s&v)z0C3F7IaLdPgSkVnYq$t*w-A1D5G8JbNeQhTlWo#r%u(y+B2E6H9LcOfRjUmXp!s&w)z0C4F7IaMdPhSkVnZq$t*x-A2D5G8KbNeQiTlWo#r%v(y+B3E6H9LcOgRjUmYp!s&w)z1C4F7JaMdPhSkWnZq$u*x-A2D5H8KbNfQiTlXo#s%v(y0B3E6I9LdOgRjVmYp!t&w)z1C4G7JaMePhSkWnZr$u*x+A2D5H8KcNfQiUlXo#s%v)y0B3F6I9LdOgSjVmYq!t&w-z1D4G7JbMePhTkWoZr$u(x+A2E5H9KcNfRiUlXp#s%v)y0C3F6IaLdOgSjVnYq!t*w-z1D4G8JbMeQhTkWoZr%u(x+B2E5H9KcOfRiUmXp#s&v)z0C3F7IaLdPgSkVnYq$t*w-A1D4G8JbNeQhTlWoZr%u(y+B2E6H9KcOfRjUmXp!s&v)z0C4F7IaMdPgSkVnZq$t*x-A1D5G8KbNeQiTlWo#r%v(y+B3E6H9LcOfRjUmYp!s&w)z0C4F7JaMdPhSkVnZq$u*x-A2D5G8KbNfQiTlXo#r%v(y0B3E6I9LcOgRjVmYp!t&w)z1C4G7JaMePhSkWnZr$u*x+A2D5H8KbNfQiUlXo#s%v(y0B3F6I9LdOgRjVmYq!t&w-z1C4G7JbMePhTkWnZr$u(x+A2E5H8KcNfRiUlXp#s%v)y0C3F6IaLdOgSjVmYq!t*w-z1D4G7JbMeQhTkWoZr$u(x+B2E5H9KcNfRiUmXp#s&v)y0C3F7IaLdPgSjVnYq$t*w-A1D4G8JbNeQhTlWoZr%u(x+B2E6H9KcOfRiUmXp!s&v)z0C3F7IaMdPgSkVnYq$t*x-A1D5G8JbNeQiTlWo#r%u(y+B3E6H9LcOfRjUmYp!s&w)z0C4F7JaMdPhSkVnZq$t*x-A2D5G8KbNeQiTlXo#r%v(y+B3E6I9LcOgRjUmYp!t&w)z1C4F7JaMePhSkWnZq$u*x+A2D5H8KbNfQiUlXo#s%v(y0B3E6I9LdOgRjVmYmYq!t&w-z1C4G7JaMePhTkWnZr$u*x+A2E5H8KcNfQiUlXp#s%v)y0B3F6IaLdOgSjVmYq!t*w-z1D4G7JbMeQhTkWoZr$u(x+B2E5H9KcNfRiUlXp#s&v)y0C3F6IaLdPgSjVnYq!t*w-A1D4G8JbMeQhTlWoZr%u(x+B2E6H9KcOfRiUmXp!s&v)z0C3F7IaMdPgSkVnYq$t*w-A1D5G8JbNeQhTlWo#r%u(y+B2E6H9LcOfRjUmXp!s&w)z0C4F7IaMdPhSkVnZq$t*x-A2D5G8KbNeQiTlXo#r%v(y+B3E6I9LcOgRjUmYp!s&w)z1C4F7JaMdPhSkWnZq$u*x-A2D5H8KbNfQiTlXo#s%v(y0B3E6I9LdOgRjVmYp!t&w-z1C4G7JaMePhTkWnZr$u*x+A2D5H8KcNfQiUlXo#s%v)y0B3F6I9LdOgSjVmYq!t&w-z1D4G7JbMePhTkWoZr$u(x+A2E5H9KcNfRiUlXp#s&v)y0C3F6IaLdOgSjVnYq!t*w-z1D4G8JbMeQhTkWoZr%u(x+B2E5H9KcOfRiUmXp#s&v)z0C3F7IaLdPgSkVnYq$t*w-A1D5G8JbNeQhTlWo#r%u(y+B2E6H9KcOfRjUmXp!s&v)z0C4F7IaMdPgSkVnZq$t*x-A1D5G8KbNeQiTlWo#r%v(y+B3E6H9LcOgRjUmYp!s&s&w)z1C4F7JaMdPhSkWnZq$u*x*x-A2D5H8KbNfQiTlXo#s%v(y0B3E6I9LcOgRjVmYp!t&w)z1C4G7JaMePhSkWnZr$u*x+A2D5H8KcNfQiUlXo#s%v)y0B3F6I9LdOgSjVmYq!t&w-z1D4G7JbMePhTkWnZr$u(x+A2E5H8KcNfRiUlXp#s%v)y0C3F6IaLdOgSjVnYq!t*w-z1D4G8JbMeQhTkWoZr%u(x+B2E5H9KcOfRiUmXp#s&v)z0C3F7IaLdPgSjVnYq$t*w-A1D4G8JbNeQhTlWoZr%u(y+B2E6H9KcOfRjUmXp!s&v)z0C4F7IaMdPgSkVnZq$t*x-A1D5G8KbNeQiTlWo#r%u(y+B3E6H9LcOfRjUmYp!s&w)z0C4F7JaMdPhSkVnZq$u*x-A2D5G8KbNfQiTlXo#r%v(y0B3E6I9LcOgRjVmYp!t&w)z1C4G7JaMePhSkWnZq$u*x+A2D5H8KbNfQiUlXo#s%v(y0B3F6I9LdOgRjVmYq!t&w-z1C4G7JbMePhTkWnZr$u(x+A2E5H8KcNfRiUlXp#s%v)y0B3F6IaLdOgSjVmYq!t*w-z1D4G7JbMeQhTkWoZr$u(x+B2E5H9KcNfRiUmXp#s&v)y0C3F7IaLdPgSjVnYq$t*w-A1D4G8JbMeQhTlWoZr%u(x+B2E6H9KcOfRiUmXp!s&v)z0C3F7IaMdPgSkVnYq$t*x-A1D5G8JbNeQiTlWo#r%u(y+B3E6H9LcOfRjUmYp!s&w)z0C4F7IaMdPhSkVnZq$t*x-A2D5G8KbNeQiTlXo#r%v(y+B3E6I9LcOgRjUmYp!t&w)z1C4F7JaMePhSkWnZq$u*x+A2D5H8KbNfQiTlXo#s%v(y0B3E6I9LdOgRjVmYp!t&w-z1C4G7JaMePhTkWnZr$u*x+A2E5H8KcNfQiUlXp#s%v)y0B3F6IaLdOgSjVmYq!t&w-z1D4G7JbMePhTkWoZr$u(x+A2E5H9KcNfRiUlXp#s&v)y0C3F6IaLdPgSjVnYq!t*w-A1D4G8JbMeQhTlWoZr%u(x+B2E6H9KcOfRiUmXp#s&v)z0C3F7IaLdPgSkVnYq$t*w-A1D5G8JbNeQhTlWo#r%u(y+B2E6H9LcOfRjUmXp!s&w)z0C4F7IaMdPhSkVnZq$t*x-A1D5G8KbNeQiTlWo#r%v(y+B3E6H9LcOgRjUmYp!s&w)z1C4F7JaMdPhSkWnZq$u*x-A2D5H8KbNfQiTlXo#s%v(y0B3E6I9L0B3E6I9LdOgRjVmYp!t&w-z1C4G7JaMePhTkWnZr$u*x+A2E5H8KcNfQiUlXp#s%v)y0B3F6IaLdOgSjVmYq!t&w-z1D4G7JbMePhTkWoZr$u(x+A2E5H9KcNfRiUlXp#s&v)y0C3F6IaLdPgSjVnYq!t*w-A1D4G8JbMeQhTlWoZr%u(x+B2E6H9KcOfRiUmXp#s&v)z0C3F7IaLdPgSkVnYq$t*w-A1D5G8JbNeQhTlWo#r%u(y+B2E6H9LcOfRjUmXp!s&w)z0C4F7IaMdPhSkVnZq$t*x-A1D5G8KbNeQiTlWo#r%v(y+B3E6H9LcOgRjUmYp!s&w)z1C4F7JaMdPhSkWnZq$u*x-A2D5H8KbNfQiTlXo#s%v(y0B3E6I9LcOgRjVmYp!t&w)z1C4G7JaMePhSkWnZr$u*x+A2D5H8KcNfQiUlXo#s%v)y0B3F6I9LdOgSjVmYq!t&w-z1D4G7JbMePhTkWoZr$u(x+A2E5H8KcNfRiUlXp#s%v)y0C3F6IaLdOgSjVnYq!t*w-z1D4G8JbMeQhTkWoZr%u(x+B2E5H9KcOfRiUmXp#s&v)z0C3F7IaLdPgSjVnYq$t*w-A1D4G8JbNeQhTlWoZr%u(y+B2E6H9KcOfRjUmXp!s&v)z0C4F7IaMdPgSkVnZq$t*x-A1D5G8KbNeQiTlWo#r%v(y+B3E6H9LcOfRjUmYp!s&w)z0C4F7JaMdPhSkVnZq$u*x-A2D5G8KbNfQiTlXo#r%v(y0B3E6I9LcOgRjVmYp!t&w)z1C4G7JaMePhSkWnZq$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