大学物理课程中“高等数学基础”教学——以应用为导向

THE TEACHING OF ADVANCED MATHEMATICS FOUNDATION IN COLLEGE PHYSICS——TAKING APPLICATIONS AS THE ORIENTATION

本文给出了为大学物理课程学习做准备的数学基础的教学中的3个"应用性"原则,指出教学重点应放在让学生理解微分与导数、积分以及矢量运算几个方面的思想与方法,特别要让学生理解:微分表示变量的微小变化;导数在几何上表示函数曲线上切线的斜率,而且导数中分子和分母两个微分是既相关联又可以分开的量;积分的本质意义是求和,把物理问题转化为积分问题通常包括两个过程,一是积分区域无限细分,其目的在于"化变为不变";二是无限求和.通过不定积分与定积分的关系可以帮助学生快速计算一些简单被积函数的积分.讲述积分的过程中,可以引入微分方程的概念及分离变量法求解的思想.矢量运算中的难点是标量积与矢量积的区别以及矢量求导.前者可通过力做功及力矩的概念帮助学生加深理解.而矢量求导可以结合曲线运动中的速度、加速度的计算让学生理解其思想与方法.

Three applicability principles are given for the teaching of mathematical foundation, which is preparative for college physics. Its keystone must be putted on letting the students understand the ideologies and methods of differential coefficient, integration, and vector oper- ations. Specially, the following views must be let students to know. Namely, differential co- efficient expresses the tiny change of variable. On geometry, a derivation is the slope of the tangent line for a curve. The numerator and denominator of a derivation are both associative and separable. Integrating is substantially equivalent to sum. It contains two processes for translating a physical problem into integration, which are fractionizing and sum illimitably. In case of the function integrated is simple, the relation between definite integration and indefi- nite integral will be helpful to students. When telling about integration, the ideas of differen- tial equation and variables separation can be introduced. The difficulties of vector operation in- clude the difference between scalar product and vector product, and the derivation of vector. The concepts of work and moment can be used to understand the former difficulty. Calculation of velocity and acceleration in curvilinear motion can be used to let students understand the thought and technique of derivation of vector.