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所在平台: Udemy |
课程主页: https://www.udemy.com/course/vector-calculus-line-surface-space-integrals-part-1/
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**课程名称：** 向量微积分（二）：线积分与曲面积分（一） **课程概述：** 本课程深入探讨向量微积分，特别是向量场中的微分和积分运算，侧重于三维欧几里得空间。课程将详细介绍线积分和曲面积分，并辅以丰富的实例。 **核心内容：** **一、线积分** * **概念引入：** 介绍线积分的基本概念，重点讲解切线方向线积分。 * **向量环量：** 深入理解向量沿曲线的环量。 * **积分计算：** * 对抛物线、连接两点的曲线进行线积分计算。 * 沿参数化曲线、连接两点的直线以及闭合路径（如三角形、矩形）进行线积分计算。 **二、曲面积分** * **概念引入：** 介绍曲面积分及法向曲面积分。 * **方向余弦与法向单位向量：** 详细讲解方向余弦和法向单位向量的概念。 * **积分计算：** * 计算曲面积分和体积分，包含详细概念讲解。 * 针对球体（以原点为中心、第一卦限内的部分球体）进行曲面积分计算。 * 针对平面（第一卦限）和圆柱体（第一卦限）进行曲面积分计算。 * **应用：** * 计算粒子在力场中移动的总功。 * 探究向量点函数是否为梯度向量。 **三、高斯散度定理** * **定理介绍：** 陈述并证明高斯散度定理及其推论，包括梯度、散度和旋度的重要结果。 * **定理应用：** 利用高斯散度定理评估曲面积分，包含带解习题。 **整体涵盖：** 课程将全面覆盖上述内容相关的所有预期定理、结果和习题。
VECTOR CALCULUS in Mathematics is a sub-division of Calculus that deals with the differentiation and integration of Vector Functions. Vector Calculus, also known as Vector Analysis, deals with the differentiation and integration of vector field, especially in the three-dimensional Euclidean space.This Course on Vector Calculus is covering the LINE INTEGRALS & SURFACE INTEGRALS that is beautifully covered with the contents that are listed below_LINE INTEGRALS1) Introduction to Line integrals and Detailed concept of Tangential Line Integrals2) Circulation of a vector along the Curves.3)Evaluating the Line integrals along the parabolic curve, curve joining the two points.4) Evaluating the Line Integrals along the Triangles, Rectangles along the path with parametric equations, along the Straight Line joining the points and along the Closed Path.SURFACE INTEGRALS1) Introduction to Surface Integrals and Normal Surface Integrals2) Detailed concepts about the Direction Cosines and Unit Vector Normal3) Evaluating the Surface Integrals & Volume Integrals with detailed concepts.4) Evaluating the Surface Integrals for a Sphere with origin at center, Sphere in the first Octant, part of the sphere in the first octant.5)Evaluating the Surface Integrals for a Plane in the first Octant, for a Cylinder in the first Octant.6) Finding the Total Work Done in moving a particle in a force Field.7) Investigating a Vector Point Function to be the Gradient Vector.GAUSS DIVERGENCE THEOREM1) Statement and Proof of Gauss Divergence Theorem and its Corollary with Important Results covering Gradient, Divergence and Curl of a Vector Function.2) Evaluating the Surface Integrals using Gauss Divergence Theorem including Solved Assignments.And, all the Expected Theorems, Results and assignments covering all the above Contents.Thanks and Regards!
