两类曲线积分的转换

1from manim import *2import numpy as np3 4config.tex_template = TexTemplateLibrary.ctex5config.tex_template.add_to_preamble(r"\setCJKmainfont{STSong}")6 7 8class CurveIntegralConversion(Scene):9    def construct(self):10        # ========== 标题 ==========11        title = Text('两类曲线积分的转换', font="SimSun", color=YELLOW).scale(0.65)12        title.to_edge(UP, buff=0.3)13        self.play(Write(title), run_time=1)14        self.wait(0.3)15 16        # =============================================17        # 场景1：问题引入——为什么要转换？18        # =============================================19        # 曲线（左到右，波浪形，保证切线角α为正锐角）20        curve_func = lambda t: np.array([21            -2.0 + 1.5 * t,22            0.8 * np.sin(1.8 * t) - 0.3,23            024        ])25        curve = ParametricFunction(curve_func, t_range=[0.2, 2.5], color=WHITE, stroke_width=3)26        self.play(Create(curve), run_time=1.5)27 28        # 起终点29        start_dot = Dot(curve_func(0.2), color=GREEN, radius=0.07)30        end_dot = Dot(curve_func(2.5), color=RED, radius=0.07)31        A_label = MathTex("A", color=GREEN).scale(0.35).next_to(start_dot, DL, buff=0.05)32        B_label = MathTex("B", color=RED).scale(0.35).next_to(end_dot, DR, buff=0.05)33        self.play(FadeIn(start_dot), FadeIn(end_dot), Write(A_label), Write(B_label), run_time=0.6)34 35        # 向量场 F=(P,Q)36        def F_field(pos):37            x, y = pos[0], pos[1]38            return np.array([0.4 * y + 0.3, -0.3 * x + 0.4, 0])39 40        field_arrows = VGroup()41        for t_val in np.linspace(0.3, 2.4, 10):42            pos = curve_func(t_val)43            F_val = F_field(pos)44            F_norm = np.linalg.norm(F_val)45            if F_norm > 0.01:46                F_dir = F_val / F_norm * 0.547                arr = Arrow(pos, pos + F_dir, buff=0, color=BLUE_C, stroke_width=2.5, tip_length=0.08)48                field_arrows.add(arr)49        self.play(LaggedStartMap(Create, field_arrows, lag_ratio=0.06), run_time=1.2)50 51        f_label = MathTex(r"\vec{F}=(P,Q)", color=BLUE_C).scale(0.38)52        f_label.next_to(curve, UP, buff=0.3).shift(RIGHT * 0.5)53        self.play(Write(f_label), run_time=0.5)54 55        # 问题：计算第二型积分56        problem_eq = MathTex(r"\text{求：}\ \int_L P\,dx + Q\,dy\ = \ ?", color=BLUE).scale(0.45)57        problem_eq.to_edge(LEFT, buff=0.4).shift(UP * 2.5)58        self.play(Write(problem_eq), run_time=1)59        self.wait(0.8)60 61        # 动机文字62        motive = Text('能否转化为对弧长的积分？', font="SimSun", color=WHITE).scale(0.3)63        motive.next_to(problem_eq, RIGHT, buff=0.3)64        self.play(Write(motive), run_time=0.8)65        self.wait(1.5)66 67        # 清理场景1（保留曲线、起终点）68        self.play(69            FadeOut(field_arrows), FadeOut(f_label),70            FadeOut(problem_eq), FadeOut(motive),71            run_time=0.6,72        )73 74        # =============================================75        # 场景2：关键洞察——切向量76        # =============================================77        scene2_label = Text('关键：切向量 T', font="SimSun", color=ORANGE).scale(0.3)78        scene2_label.to_edge(LEFT, buff=0.4).shift(UP * 2.5)79        self.play(Write(scene2_label), run_time=0.5)80 81        # 动态切向量82        def get_tangent_at(t_val):83            dt = 0.0184            p1 = curve_func(t_val)85            p2 = curve_func(t_val + dt)86            T = (p2 - p1) / np.linalg.norm(p2 - p1)87            return T88 89        t_tracker = ValueTracker(0.5)90        moving_point = always_redraw(91            lambda: Dot(curve_func(t_tracker.get_value()), color=ORANGE, radius=0.08)92        )93        tangent_arrow = always_redraw(94            lambda: Arrow(95                curve_func(t_tracker.get_value()),96                curve_func(t_tracker.get_value()) + get_tangent_at(t_tracker.get_value()) * 1.0,97                buff=0, color=ORANGE, stroke_width=3.5, tip_length=0.12,98            )99        )100        t_label_dyn = always_redraw(101            lambda: MathTex(r"\vec{T}", color=ORANGE).scale(0.38).next_to(102                curve_func(t_tracker.get_value()) + get_tangent_at(t_tracker.get_value()) * 1.0,103                UR, buff=0.03,104            )105        )106 107        self.play(FadeIn(moving_point), Create(tangent_arrow), Write(t_label_dyn), run_time=0.6)108        self.play(t_tracker.animate.set_value(0.5), run_time=2.0, rate_func=linear)109        self.wait(0.3)110 111        # 冻结在 t=0.5（α≈30°，正锐角，位置清晰）112        t_stop = t_tracker.get_value()113        P_pos = curve_func(t_stop)114        T_vec = get_tangent_at(t_stop)115        self.remove(moving_point, tangent_arrow, t_label_dyn)116 117        static_point = Dot(P_pos, color=ORANGE, radius=0.08)118        static_tangent = Arrow(119            P_pos, P_pos + T_vec * 1.0,120            buff=0, color=ORANGE, stroke_width=3.5, tip_length=0.12,121        )122        static_t_label = MathTex(r"\vec{T}", color=ORANGE).scale(0.38)123        static_t_label.next_to(static_tangent.get_end(), UR, buff=0.03)124        self.add(static_point, static_tangent, static_t_label)125 126        # x轴参考线 + 角度127        x_ref = DashedLine(P_pos + LEFT * 0.2, P_pos + RIGHT * 1.3, color=GREY, stroke_width=1.5)128        self.play(Create(x_ref), run_time=0.4)129 130        alpha_val = np.arctan2(T_vec[1], T_vec[0])131        angle_arc = Angle(132            Line(P_pos, P_pos + RIGHT), Line(P_pos, P_pos + T_vec),133            radius=0.3, color=RED_C,134        )135        alpha_label = MathTex(r"\alpha", color=RED_C).scale(0.35)136        alpha_label.move_to(P_pos + 0.45 * np.array([np.cos(alpha_val / 2), np.sin(alpha_val / 2), 0]))137        self.play(Create(angle_arc), Write(alpha_label), run_time=0.8)138 139        # 分解为直角三角形140        dx_comp = np.array([T_vec[0], 0, 0]) * 1.0141        dy_comp = np.array([0, T_vec[1], 0]) * 1.0142 143        h_line = DashedLine(P_pos, P_pos + dx_comp, color=RED_C, stroke_width=2.5)144        v_line = DashedLine(P_pos + dx_comp, P_pos + dx_comp + dy_comp, color=TEAL, stroke_width=2.5)145        self.play(Create(h_line), Create(v_line), run_time=0.8)146 147        # 标注分量148        cos_a_label = MathTex(r"\cos\alpha", color=RED_C).scale(0.3)149        cos_a_label.next_to(h_line, DOWN, buff=0.05)150        cos_b_label = MathTex(r"\cos\beta", color=TEAL).scale(0.3)151        cos_b_label.next_to(v_line, RIGHT, buff=0.05)152        self.play(Write(cos_a_label), Write(cos_b_label), run_time=0.6)153 154        # 右侧公式155        formula_T = MathTex(r"\vec{T} = (\cos\alpha,\ \cos\beta)", color=ORANGE).scale(0.4)156        formula_T.to_edge(RIGHT, buff=0.4).shift(UP * 1.8)157        self.play(Write(formula_T), run_time=0.8)158        self.wait(1.5)159 160        self.play(FadeOut(scene2_label), run_time=0.3)161 162        # =============================================163        # 场景3：微元关系 ds, dx, dy164        # =============================================165        scene3_label = Text('微元关系：ds → dx, dy', font="SimSun", color=WHITE).scale(0.28)166        scene3_label.to_edge(LEFT, buff=0.4).shift(UP * 2.5)167        self.play(Write(scene3_label), run_time=0.5)168 169        # 用高亮曲线段表示 ds（实际弧长）170        dt_small = 0.35171        arc_segment = ParametricFunction(172            curve_func, t_range=[t_stop, t_stop + dt_small],173            color=YELLOW, stroke_width=5,174        )175        arc_end = curve_func(t_stop + dt_small)176        ds_brace = BraceBetweenPoints(P_pos, arc_end, color=YELLOW)177        ds_label = MathTex("ds", color=YELLOW).scale(0.4)178        ds_label.next_to(ds_brace, ds_brace.get_direction(), buff=0.05)179        self.play(Create(arc_segment), run_time=0.6)180        self.play(Create(ds_brace), Write(ds_label), run_time=0.6)181 182        # 显示切线段≈弧长段183        tangent_end = P_pos + T_vec * np.linalg.norm(arc_end - P_pos)184        tangent_seg = Line(P_pos, tangent_end, color=ORANGE, stroke_width=4)185        approx_text = MathTex(r"\text{\color{yellow}弧长} \approx \text{\color{orange}切线段}", color=WHITE).scale(0.3)186        approx_text.next_to(ds_brace, ds_brace.get_direction(), buff=0.35)187        self.play(Create(tangent_seg), Write(approx_text), run_time=0.8)188        self.wait(1)189 190        # 分解切线段为 dx, dy191        seg_len = np.linalg.norm(arc_end - P_pos)192        dx_comp_seg = np.array([T_vec[0], 0, 0]) * seg_len193        dy_comp_seg = np.array([0, T_vec[1], 0]) * seg_len194        h_line_seg = DashedLine(P_pos, P_pos + dx_comp_seg, color=RED_C, stroke_width=2.5)195        v_line_seg = DashedLine(P_pos + dx_comp_seg, P_pos + dx_comp_seg + dy_comp_seg, color=TEAL, stroke_width=2.5)196        self.play(Create(h_line_seg), Create(v_line_seg), run_time=0.6)197 198        # 标注 dx, dy199        dx_label = MathTex("dx", color=RED_C).scale(0.3)200        dx_label.next_to(h_line_seg, DOWN, buff=0.12)201        dy_label = MathTex("dy", color=TEAL).scale(0.3)202        dy_label.next_to(v_line_seg, RIGHT, buff=0.1)203        self.play(Write(dx_label), Write(dy_label), run_time=0.5)204 205        # 局部放大：复制几何元素居中放大显示206        zoom_copy = VGroup(207            arc_segment.copy(),208            tangent_seg.copy(),209            ds_brace.copy(), ds_label.copy(),210            h_line_seg.copy(), v_line_seg.copy(),211            dx_label.copy(), dy_label.copy(),212            angle_arc.copy(), alpha_label.copy(),213            static_point.copy(),214        )215        zoom_copy.scale(2.5).move_to(ORIGIN + LEFT * 0.5)216        zoom_bg = SurroundingRectangle(zoom_copy, color=WHITE, buff=0.2, corner_radius=0.1, fill_color=BLACK, fill_opacity=0.85)217        zoom_title = Text('放大：曲线段 ds → dx, dy', font="SimSun", color=YELLOW).scale(0.3)218        zoom_title.next_to(zoom_bg, UP, buff=0.1)219        self.play(FadeIn(zoom_bg), FadeIn(zoom_copy), Write(zoom_title), run_time=1)220        self.wait(3)221        self.play(FadeOut(zoom_bg), FadeOut(zoom_copy), FadeOut(zoom_title), run_time=0.8)222 223        # 关键公式（带闪烁提示）224        key_eq1 = MathTex(r"dx = \cos\alpha \cdot ds", color=RED_C).scale(0.42)225        key_eq1.next_to(formula_T, DOWN, buff=0.25)226        key_eq2 = MathTex(r"dy = \cos\beta \cdot ds", color=TEAL).scale(0.42)227        key_eq2.next_to(key_eq1, DOWN, buff=0.1)228 229        # 闪烁 h_line 同时写公式230        self.play(Write(key_eq1), h_line.animate.set_color(YELLOW), run_time=1)231        self.play(h_line.animate.set_color(RED_C), run_time=0.3)232        self.play(Write(key_eq2), v_line.animate.set_color(YELLOW), run_time=1)233        self.play(v_line.animate.set_color(TEAL), run_time=0.3)234 235        bridge_text = Text('这就是连接两类积分的桥梁！', font="SimSun", color=YELLOW).scale(0.25)236        bridge_text.next_to(key_eq2, DOWN, buff=0.15)237        self.play(Write(bridge_text), run_time=0.8)238        self.wait(2)239 240        # =============================================241        # 场景4：代入——得到转换公式242        # =============================================243        # 清理几何元素（保留曲线）244        geom_objs = VGroup(245            static_point, static_tangent, static_t_label,246            x_ref, angle_arc, alpha_label,247            h_line, v_line, cos_a_label, cos_b_label,248            arc_segment, tangent_seg, approx_text,249            h_line_seg, v_line_seg,250            ds_brace, ds_label, dx_label, dy_label,251            scene3_label, bridge_text,252        )253        self.play(FadeOut(geom_objs), run_time=0.8)254 255        # 推导放在右侧（下移避免重叠），曲线在左侧保留256        sub_title = Text('代入 dx, dy：', font="SimSun", color=WHITE).scale(0.28)257        sub_title.to_edge(RIGHT, buff=1.2).shift(DOWN * 0.2)258        self.play(Write(sub_title), run_time=0.5)259 260        line1 = MathTex(r"\int_L", r"P\,dx", r"+", r"Q\,dy", color=BLUE).scale(0.38)261        line1.next_to(sub_title, DOWN, buff=0.1)262        self.play(Write(line1), run_time=0.8)263 264        # 高亮 dx, dy265        self.play(266            line1[1].animate.set_color(RED_C),267            line1[3].animate.set_color(TEAL),268            run_time=0.5,269        )270 271        line2 = MathTex(272            r"= \int_L P\cos\alpha\,ds + Q\cos\beta\,ds",273            color=WHITE,274        ).scale(0.34)275        line2.next_to(line1, DOWN, buff=0.08)276        self.play(Write(line2), run_time=1.2)277 278        # 最终公式（黄色高亮框，包含完整等式，左移避免出框）279        final_eq = MathTex(280            r"\int_L P\,dx + Q\,dy",281            r"=",282            r"\int_L (P\cos\alpha + Q\cos\beta)\,ds",283            color=WHITE,284        ).scale(0.36)285        final_eq[0].set_color(BLUE)286        final_eq[2].set_color(YELLOW)287        final_eq.next_to(line2, DOWN, buff=0.12).shift(LEFT * 0.3)288        final_box = SurroundingRectangle(final_eq, color=YELLOW, buff=0.08, corner_radius=0.05)289        self.play(Write(final_eq), Create(final_box), run_time=1.5)290        self.wait(2)291 292        # 清理推导细节，保留最终公式293        deriv_cleanup = VGroup(formula_T, key_eq1, key_eq2, sub_title, line1, line2)294        self.play(FadeOut(deriv_cleanup), run_time=0.6)295 296        # 将最终公式移到上方保留297        self.play(298            final_eq.animate.scale(0.9).next_to(title, DOWN, buff=0.2),299            final_box.animate.scale(0.9).next_to(title, DOWN, buff=0.2),300            run_time=1,301        )302        new_box = SurroundingRectangle(final_eq, color=YELLOW, buff=0.08, corner_radius=0.05)303        self.play(Transform(final_box, new_box), run_time=0.3)304 305        # =============================================306        # 场景5：几何意义——向量场在切线方向的投影307        # =============================================308        # 重新显示向量场309        field_arrows2 = VGroup()310        for t_val in np.linspace(0.3, 2.4, 10):311            pos = curve_func(t_val)312            F_val = F_field(pos)313            F_norm = np.linalg.norm(F_val)314            if F_norm > 0.01:315                F_dir = F_val / F_norm * 0.5316                arr = Arrow(pos, pos + F_dir, buff=0, color=BLUE_C, stroke_width=2, tip_length=0.07)317                field_arrows2.add(arr)318        self.play(LaggedStartMap(Create, field_arrows2, lag_ratio=0.05), run_time=1.2)319 320        # 明确说明：被积函数 Pcosα+Qcosβ 的几何意义321        geo_title = MathTex(322            r"P\cos\alpha + Q\cos\beta",323            r"\ = \ ",324            r"\vec{F}\cdot\vec{T}",325            r"\ = \ ",326            r"\text{向量场在切线上的投影}",327            color=WHITE,328        ).scale(0.32)329        geo_title[0].set_color(YELLOW)330        geo_title[2].set_color(ORANGE)331        geo_title[4].set_color(YELLOW)332        geo_title.to_edge(DOWN, buff=0.5)333        self.play(Write(geo_title), run_time=1)334 335        # 动态投影演示336        proj_tracker = ValueTracker(0.5)337 338        demo_dot = always_redraw(339            lambda: Dot(curve_func(proj_tracker.get_value()), color=WHITE, radius=0.07)340        )341 342        def make_proj_visual():343            t_val = proj_tracker.get_value()344            pos = curve_func(t_val)345            T = get_tangent_at(t_val)346            F_val = F_field(pos)347            F_norm = np.linalg.norm(F_val)348            if F_norm < 0.01:349                return VGroup()350            F_dir = F_val / F_norm * 0.65351            f_arr = Arrow(pos, pos + F_dir, buff=0, color=BLUE, stroke_width=3, tip_length=0.1)352            t_arr = Arrow(pos, pos + T * 0.65, buff=0, color=ORANGE, stroke_width=3, tip_length=0.1)353            proj_len = np.dot(F_dir, T)354            proj_end = pos + T * proj_len355            proj_arr = Arrow(pos, proj_end, buff=0, color=YELLOW, stroke_width=3.5, tip_length=0.09)356            drop = DashedLine(pos + F_dir, proj_end, color=GREY_B, stroke_width=1.5)357            return VGroup(f_arr, t_arr, proj_arr, drop)358 359        proj_visual = always_redraw(make_proj_visual)360        self.add(demo_dot, proj_visual)361        self.wait(0.3)362 363        self.play(proj_tracker.animate.set_value(2.5), run_time=5, rate_func=linear)364        self.wait(1)365 366        # 清理367        self.remove(demo_dot, proj_visual)368        self.play(FadeOut(field_arrows2), FadeOut(geo_title), run_time=0.6)369 370        # =============================================371        # 场景6：方向性备注（配合切向量动画演示）372        # =============================================373        # 左侧：重新显示曲线 + 切向量动画演示方向反转374        dir_title = Text('补充：方向性', font="SimSun", color=ORANGE).scale(0.35)375        dir_title.to_edge(LEFT, buff=0.4).shift(UP * 2.5)376        self.play(Write(dir_title), run_time=0.5)377 378        # 正向切向量（锐角α）379        t_demo = 0.5380        P_demo = curve_func(t_demo)381        T_fwd = get_tangent_at(t_demo)382        fwd_arrow = Arrow(P_demo, P_demo + T_fwd * 1.2, buff=0, color=GREEN, stroke_width=4, tip_length=0.12)383        fwd_label = MathTex(r"\vec{T}", color=GREEN).scale(0.4).next_to(fwd_arrow.get_end(), UR, buff=0.05)384 385        # x轴参考线 + 角度386        x_ref_dir = DashedLine(P_demo + LEFT * 0.3, P_demo + RIGHT * 1.5, color=GREY, stroke_width=1.5)387        alpha_fwd = np.arctan2(T_fwd[1], T_fwd[0])388        arc_fwd = Angle(389            Line(P_demo, P_demo + RIGHT), Line(P_demo, P_demo + T_fwd),390            radius=0.35, color=GREEN,391        )392        alpha_fwd_label = MathTex(r"\alpha", color=GREEN).scale(0.35)393        alpha_fwd_label.move_to(P_demo + 0.55 * np.array([np.cos(alpha_fwd / 2), np.sin(alpha_fwd / 2), 0]))394 395        dot_demo = Dot(P_demo, color=WHITE, radius=0.07)396        self.play(FadeIn(dot_demo), Create(x_ref_dir), Create(fwd_arrow), Write(fwd_label), run_time=0.8)397        self.play(Create(arc_fwd), Write(alpha_fwd_label), run_time=0.6)398 399        # 右侧：逻辑链说明（正向）400        step0 = Text('正向行走：', font="SimSun", color=GREEN).scale(0.25)401        step0.to_edge(RIGHT, buff=1.2).shift(UP * 1.2)402        step1 = MathTex(403            r"\vec{T} \text{ 指向前方}",404            color=GREEN,405        ).scale(0.33)406        step1.next_to(step0, DOWN, buff=0.08)407        step1_alpha = MathTex(408            r"\Rightarrow \alpha \text{ 为锐角}",409            color=GREEN,410        ).scale(0.33)411        step1_alpha.next_to(step1, DOWN, buff=0.06)412        step1_cos = MathTex(413            r"\Rightarrow \cos\alpha > 0",414            color=GREEN,415        ).scale(0.33)416        step1_cos.next_to(step1_alpha, DOWN, buff=0.06)417        self.play(Write(step0), run_time=0.4)418        self.play(Write(step1), run_time=0.6)419        self.play(Write(step1_alpha), run_time=0.6)420        self.play(Write(step1_cos), run_time=0.6)421        self.wait(1.5)422 423        # 反向切向量（钝角）424        T_bwd = -T_fwd425        bwd_arrow = Arrow(P_demo, P_demo + T_bwd * 1.2, buff=0, color=RED_C, stroke_width=4, tip_length=0.12)426        bwd_label = MathTex(r"-\vec{T}", color=RED_C).scale(0.4).next_to(bwd_arrow.get_end(), DL, buff=0.05)427 428        alpha_bwd = np.arctan2(T_bwd[1], T_bwd[0])429        arc_bwd = Angle(430            Line(P_demo, P_demo + RIGHT), Line(P_demo, P_demo + T_bwd),431            radius=0.35, color=RED_C,432        )433        alpha_bwd_label = MathTex(r"\pi-\alpha", color=RED_C).scale(0.32)434        alpha_bwd_label.move_to(P_demo + 0.6 * np.array([np.cos((np.pi + alpha_bwd) / 2), np.sin((np.pi + alpha_bwd) / 2), 0]))435 436        self.play(437            Transform(fwd_arrow, bwd_arrow),438            Transform(fwd_label, bwd_label),439            Transform(arc_fwd, arc_bwd),440            Transform(alpha_fwd_label, alpha_bwd_label),441            run_time=1.5,442        )443 444        # 右侧：逻辑链说明（反向）445        step2 = Text('反向行走：', font="SimSun", color=RED_C).scale(0.25)446        step2.next_to(step1_cos, DOWN, buff=0.2)447        chain1 = MathTex(448            r"\text{方向反转}",449            r"\Rightarrow",450            r"\vec{T} \to -\vec{T}",451            color=RED_C,452        ).scale(0.33)453        chain1.next_to(step2, DOWN, buff=0.08)454        chain2 = MathTex(455            r"\Rightarrow \alpha \to \pi-\alpha \text{（变为钝角）}",456            color=RED_C,457        ).scale(0.33)458        chain2.next_to(chain1, DOWN, buff=0.06)459        chain3 = MathTex(460            r"\Rightarrow \cos\alpha < 0",461            color=RED_C,462        ).scale(0.33)463        chain3.next_to(chain2, DOWN, buff=0.06)464        self.play(Write(step2), run_time=0.4)465        self.play(Write(chain1), run_time=0.7)466        self.play(Write(chain2), run_time=0.7)467        self.play(Write(chain3), run_time=0.6)468        self.wait(1.5)469 470        # 结论471        dir_c1_note = MathTex(472            r"\therefore\quad",473            r"\int_L (P\cos\alpha+Q\cos\beta)\,ds",474            r"\text{ 变号}",475            color=BLUE,476        ).scale(0.34)477        dir_c1_note.next_to(chain3, DOWN, buff=0.15)478        self.play(Write(dir_c1_note), run_time=0.8)479        self.wait(3)480 481 482def main():483    import os484    os.system("manim -pql curve_integral_conversion.py CurveIntegralConversion")485 486 487if __name__ == "__main__":488    main()