全微分的几何意义

1# -*- coding: utf-8 -*-2from manim import *3import numpy as np4 5class TotalDifferential(ThreeDScene):6    def construct(self):7        # 创建3D坐标系并向右下方移动，增加刻度间距8        axes = ThreeDAxes(9            x_range=[-2, 2, 0.5],  # 增加刻度间距10            y_range=[-2, 2, 0.5],11            z_range=[-1, 3, 0.5],12            x_length=12,  # 增加坐标轴长度以放大图像13            y_length=12,14            z_length=815        ).shift(2 * DOWN + RIGHT)  # 向右下方移动更多16        17        # 设置初始相机角度18        self.set_camera_orientation(phi=60 * DEGREES, theta=-45 * DEGREES)19 20        # 添加坐标轴标签21        x_label = MathTex("x").next_to(axes.x_axis.get_end(), RIGHT)22        y_label = MathTex("y").next_to(axes.y_axis.get_end(), UP)23        z_label = MathTex("z").next_to(axes.z_axis.get_end(), OUT)24        labels = VGroup(x_label, y_label, z_label)25 26        # 创建函数z = x^2 + y^2的曲面，缩小范围27        def func(x, y):28            return x**2 + y**229 30        surface = Surface(31            lambda u, v: axes.c2p(u, v, func(u, v)),32            u_range=[-1.5, 1.5],  # 缩小范围33            v_range=[-1.5, 1.5],34            resolution=(50, 50),  # 增加网格线密度35            fill_color=PURPLE,  # 更改曲面颜色为紫色36            fill_opacity=0.6  # 设置不透明度为0.637        )38 39        # 显示坐标系和曲面40        self.play(Create(axes), Write(labels))41        self.play(Create(surface))42        self.wait(1)43 44        # 选择一个点(x,y)并显示45        x0, y0 = 0.5, 0.546        point = Dot3D(axes.c2p(x0, y0, func(x0, y0)), color=YELLOW, radius=0.1)47        self.play(Create(point))48 49        # 创建垂直于z轴的水平平面并显示50        horizontal_plane = Surface(51            lambda u, v: axes.c2p(u, v, func(x0, y0)),52            u_range=[-1.5, 1.5],53            v_range=[-1.5, 1.5],54            resolution=(20, 20),  # 增加网格线密度55            fill_opacity=0.3,  # 设置不透明度为0.356            color=BLUE57        )58        self.play(Create(horizontal_plane))59 60        # 在水平面上选取新点并创建垂线61        dx, dy = 0.3, 0.362        new_point = axes.c2p(x0 + dx, y0 + dy, func(x0, y0))63        new_dot = Dot3D(new_point, color=WHITE, radius=0.05)64        self.play(Create(new_dot))65 66        # 垂线与实际增量相交67        actual_intersection = axes.c2p(x0 + dx, y0 + dy, func(x0 + dx, y0 + dy))68        actual_line = DashedLine(new_point, actual_intersection, color=YELLOW_D)69        self.play(Create(actual_line))70 71        # 创建切平面72        def get_tangent_plane(x0, y0, delta=1):73            # 计算偏导数74            fx = 2 * x075            fy = 2 * y076            f_val = func(x0, y0)77            78            # 切平面方程：z = f(x0,y0) + fx(x-x0) + fy(y-y0)79            def plane_func(x, y):80                return f_val + fx*(x-x0) + fy*(y-y0)81            82            plane = Surface(83                lambda u, v: axes.c2p(u, v, plane_func(u, v)),84                u_range=[x0-delta, x0+delta],85                v_range=[y0-delta, y0+delta],86                resolution=(40, 40),  # 增加网格线密度87                fill_opacity=0.6,88                color=ORANGE  # 更改切平面颜色为橙色89            )90            return plane91 92        # 创建并显示切平面93        tangent_plane = get_tangent_plane(x0, y0)94        self.play(Create(tangent_plane))95 96        # 垂线与线性近似相交97        approx_intersection = axes.c2p(x0 + dx, y0 + dy, func(x0, y0) + 2*x0*dx + 2*y0*dy)98        approx_line = DashedLine(new_point, approx_intersection, color=GREEN_D)99        self.play(Create(approx_line))100 101        # 显示误差线102        error_line = Line(actual_intersection, approx_intersection, color=RED_B, stroke_width=2)103        self.play(Create(error_line))104 105        # 添加文字说明106        explanation_text = VGroup(107            Text("点(x0+Δx, y0+Δy)到曲面距离为Δz", font="SimSun", color=YELLOW_D).scale(0.4),108            Text("点(x0+Δx, y0+Δy)到切平面距离为dz", font="SimSun", color=GREEN_D).scale(0.4),109           # Text("切平面到曲面的距离为误差", font="SimSun", color=RED_B).scale(0.4)110        ).arrange(DOWN, aligned_edge=LEFT)111        explanation_text.to_corner(DR)  # 移动到右下角112        self.add_fixed_in_frame_mobjects(explanation_text)113        self.play(Write(explanation_text))114 115        # 调整相机运动116        self.move_camera(phi=70 * DEGREES, theta=-30 * DEGREES, run_time=2)117        self.begin_ambient_camera_rotation(rate=0.15)118        self.wait(6)119        self.stop_ambient_camera_rotation()120        121        # 最后移动到另一个角度以更好地观察122        self.move_camera(phi=45 * DEGREES, theta=60 * DEGREES, run_time=2)123        self.wait(2)