月更 技术博客 CXX优先 Zola 2026-08-11T00:00:00+00:00 https://suchaharcan.github.io/tags/multivariable/atom.xml 2026-08-11T00:00:00+00:00 2026-08-11T00:00:00+00:00 Suchaharcan https://suchaharcan.github.io/00Read_blog_Math/math-calculus-multi/ <h1 id="gao-deng-shu-xue-duo-yuan-wei-ji-fen">高等数学 · 多元微积分</h1> <blockquote> <p>考研高数后半程：<strong>偏导 → 全微分 → 方向导数/梯度 → 极值 → 重积分 → 线面积分 → 三大公式</strong>。</p> </blockquote> <hr /> <h2 id="1-duo-yuan-han-shu-ji-xian-yu-lian-xu">1. 多元函数极限与连续</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>lim((x,y)→(x₀,y₀)) f(x,y) = A</span></span></code></pre> <p><strong>判断极限不存在</strong>：取两条不同路径（如 y=kx、y=x²），极限不同则不存在。</p> <hr /> <h2 id="2-pian-dao-shu">2. 偏导数</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>fₓ(x₀,y₀) = lim(Δx→0) [f(x₀+Δx, y₀) − f(x₀,y₀)] / Δx</span></span> <span class="giallo-l"><span>fᵧ(x₀,y₀) = lim(Δy→0) [f(x₀, y₀+Δy) − f(x₀,y₀)] / Δy</span></span></code></pre><h3 id="gao-jie-pian-dao">高阶偏导</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>fₓₓ、fₓᵧ、fᵧₓ、fᵧᵧ</span></span> <span class="giallo-l"><span>若 fₓᵧ 与 fᵧₓ 连续 ⇒ 二者相等（混合偏导与次序无关）</span></span></code></pre><h3 id="yin-han-shu-pian-dao">隐函数偏导</h3> <p><code>F(x,y,z)=0</code> 确定 <code>z=z(x,y)</code>：</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∂z/∂x = −Fₓ/F_z，∂z/∂y = −Fᵧ/F_z</span></span></code></pre> <hr /> <h2 id="3-quan-wei-fen">3. 全微分</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>dz = fₓ(x,y)·dx + fᵧ(x,y)·dy</span></span></code></pre> <p><strong>可微 ⇒ 偏导存在；偏导连续 ⇒ 可微；可微 ⇒ 连续。</strong></p> <p>复合函数链式法则（中间变量法）：</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>z = f(u,v)，u=u(x,y)，v=v(x,y)</span></span> <span class="giallo-l"><span>∂z/∂x = (∂z/∂u)(∂u/∂x) + (∂z/∂v)(∂v/∂x)</span></span></code></pre> <hr /> <h2 id="4-fang-xiang-dao-shu-yu-ti-du">4. 方向导数与梯度</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>方向导数 ∂f/∂l = fₓ·cos α + fᵧ·cos β</span></span> <span class="giallo-l"><span>梯度 grad f = (fₓ, fᵧ)</span></span></code></pre> <p><strong>梯度方向是函数增长最快的方向</strong>，其模为方向导数最大值。</p> <hr /> <h2 id="5-duo-yuan-han-shu-ji-zhi">5. 多元函数极值</h2> <h3 id="wu-tiao-jian-ji-zhi-er-jie-pian-dao-pan-bie">无条件极值（二阶偏导判别）</h3> <p>驻点 <code>(x₀,y₀)</code>：<code>fₓ=fᵧ=0</code>，记</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>A = fₓₓ，B = fₓᵧ，C = fᵧᵧ</span></span> <span class="giallo-l"><span>AC − B² &gt; 0 且 A &lt; 0 ⇒ 极大；AC − B² &gt; 0 且 A &gt; 0 ⇒ 极小</span></span> <span class="giallo-l"><span>AC − B² &lt; 0 ⇒ 非极值（鞍点）；AC − B² = 0 ⇒ 需另判</span></span></code></pre><h3 id="tiao-jian-ji-zhi-la-ge-lang-ri-cheng-shu-fa">条件极值（拉格朗日乘数法）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>在 φ(x,y)=0 下求 f(x,y) 极值：</span></span> <span class="giallo-l"><span>L(x,y,λ) = f(x,y) + λ·φ(x,y)</span></span> <span class="giallo-l"><span>解 Lₓ=Lᵧ=L_λ=0</span></span></code></pre> <hr /> <h2 id="6-er-zhong-ji-fen">6. 二重积分</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∬(D) f(x,y) dxdy</span></span></code></pre><h3 id="zhi-jiao-zuo-biao">直角坐标</h3> <table><thead><tr><th>区域类型</th><th>积分次序</th></tr></thead><tbody> <tr><td>X 型</td><td><code>∫(a..b) dx ∫(y₁(x)..y₂(x)) f dy</code></td></tr> <tr><td>Y 型</td><td><code>∫(c..d) dy ∫(x₁(y)..x₂(y)) f dx</code></td></tr> </tbody></table> <h3 id="ji-zuo-biao">极坐标</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>x = r·cos θ，y = r·sin θ，dxdy = r·drdθ</span></span></code></pre> <p>适合圆域、被积函数含 <code>x²+y²</code>。</p> <h3 id="dui-cheng-xing">对称性</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>D 关于 x 轴对称：f 关于 y 为奇 ⇒ 0；为偶 ⇒ 2 倍</span></span> <span class="giallo-l"><span>D 关于 y=x 对称：∬f(x,y) = ∬f(y,x)</span></span></code></pre> <hr /> <h2 id="7-san-zhong-ji-fen">7. 三重积分</h2> <table><thead><tr><th>方法</th><th>适用</th><th>公式</th></tr></thead><tbody> <tr><td>先一后二（投影法）</td><td>一般区域</td><td><code>∫∫ dxdy ∫(z₁..z₂) f dz</code></td></tr> <tr><td>先二后一（截面法）</td><td>旋转体</td><td><code>∫(a..b) dz ∬(D_z) f dxdy</code></td></tr> <tr><td>柱坐标</td><td>旋转体</td><td><code>x=r cosθ, y=r sinθ, z=z</code>，dv = r drdθdz</td></tr> <tr><td>球坐标</td><td>球域</td><td><code>x=r sinφ cosθ, …</code>，dv = r² sinφ drdφdθ</td></tr> </tbody></table> <hr /> <h2 id="8-qu-xian-ji-fen">8. 曲线积分</h2> <h3 id="di-yi-lei-dui-hu-chang">第一类（对弧长）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∫(L) f(x,y) ds，参数化 x=x(t),y=y(t)：</span></span> <span class="giallo-l"><span>ds = √(x&#39;² + y&#39;²) dt</span></span></code></pre> <p>与方向无关，物理意义：质量、弧长。</p> <h3 id="di-er-lei-dui-zuo-biao">第二类（对坐标）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∫(L) P dx + Q dy = ∫ P·x&#39;(t) dt + Q·y&#39;(t) dt</span></span></code></pre> <p>与方向有关。</p> <h3 id="ge-lin-gong-shi">格林公式</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∮(L⁺) P dx + Q dy = ∬(D) (∂Q/∂x − ∂P/∂y) dxdy</span></span></code></pre> <p><strong>与路径无关 ⇔ ∂P/∂y = ∂Q/∂x ⇔ 存在势函数 u 使 du = P dx + Q dy</strong>。</p> <hr /> <h2 id="9-qu-mian-ji-fen">9. 曲面积分</h2> <h3 id="di-yi-lei-dui-mian-ji">第一类（对面积）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∬(Σ) f dS = ∬(D) f(x,y,z(x,y))·√(1+zₓ²+zᵧ²) dxdy</span></span></code></pre><h3 id="di-er-lei-dui-zuo-biao-1">第二类（对坐标）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∬(Σ) P dydz + Q dzdx + R dxdy，定向曲面，与侧有关</span></span></code></pre><h3 id="gao-si-gong-shi-tong-liang">高斯公式（通量）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∯(Σ) P dydz + Q dzdx + R dxdy = ∭(Ω) (∂P/∂x + ∂Q/∂y + ∂R/∂z) dv</span></span></code></pre><h3 id="si-tuo-ke-si-gong-shi-huan-liang">斯托克斯公式（环量）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>∮(L) P dx + Q dy + R dz = ∬(Σ) rot F · n dS</span></span></code></pre> <hr /> <h2 id="10-kao-yan-chang-kao-ti-xing">10. 考研常考题型</h2> <ol> <li>二重积分换序 / 极坐标转换；</li> <li>对称性化简重积分；</li> <li>条件极值（拉格朗日）应用题；</li> <li>格林公式补线（挖洞：奇点处小圆）；</li> <li>高斯公式补面、方向（内侧/外侧）；</li> <li>方向导数与梯度求最值方向。</li> </ol> <h3 id="li-ti-su-lan">例题速览</h3> <p><strong>例 1</strong>：<code>∬(D) (x²+y²) dxdy</code>，D 为 <code>x²+y² ≤ 1</code> 第一象限。</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>极坐标：∫(0..π/2) dθ ∫(0..1) r²·r dr = (π/2)·(1/4) = π/8</span></span></code></pre> <p><strong>例 2</strong>：求 <code>f(x,y)=x²+y²</code> 在 <code>x+y=1</code> 下的最小值。</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>L = x²+y² + λ(x+y−1)</span></span> <span class="giallo-l"><span>2x+λ=0，2y+λ=0，x+y=1 ⇒ x=y=1/2</span></span> <span class="giallo-l"><span>最小值 = 1/4 + 1/4 = 1/2</span></span></code></pre> <hr /> <h2 id="11-yi-cuo-dian-qing-dan">11. 易错点清单</h2> <ul> <li>混合偏导相等需先验证连续；</li> <li>二重积分换序必画区域图；</li> <li>极坐标别漏因子 <code>r</code>；</li> <li>第二类曲线/曲面积分注意方向与侧；</li> <li>格林/高斯公式使用前检查封闭性与奇点。</li> </ul> <blockquote> <p>上一篇：<a href="https://suchaharcan.github.io/00Read_blog_Math/math-calculus-one/">一元微积分</a> ｜ 下一篇：<a href="https://suchaharcan.github.io/00Read_blog_Math/math-linear-algebra/">线性代数</a></p> </blockquote>