We begin studying ordinary differential equations by deriving them from physical or other problems (modeling), solving them by standard methods, and interpreting solutions and graphs in terms of a given problem.
The simplest ordinary differential equations, called ODEs of the first order are to be initiated. In the second chapter we discuss linear ODEs of the second order. This chapter includes the derivation of general and particular solutions. Then we extend the concepts and methods for linear ODEs to orders more than 3.
In chapter 6 we consider the Laplace transform and its application to engineering problems involving ODEs. The Laplace transform is a powerful method for solving linear ODEs and corresponding initial value problems without first determining a general a general solution.

It is expected that the students be acquainted with solid knowledge of basic principles, methods, and results, and a clear view what engineering mathematics is all about, and that it requires proficiency in all three phases of problem solving:
• Modeling, that is, translating a physical or other problem into a mathematical forms, into a mathematical model; this can be an algebraic equation, a differential equation, a graph, or some other mathematical expression.
• Solving the model by selecting and applying a suitable mathematical method, often requiring numeric work on a computer.
• Interpreting the mathematical result in physical or other terms to see what it practically means and implies.

9.1二度及三度空間向量
9.2內積
9.3向量積
9.4向量場與純量場‧導函數
9.5曲線‧弧長‧曲率‧ 轉距
9.6多變數函數
9.7純量場梯度‧方向導函數
9.8向量場散度
9.9向量場旋度

10.1 線積分
10.2 與路境無關之線積分
10.3 雙重積分
10.4 平面之葛林定理
10.6 面積分
10.7 三重積分‧高斯發散定理
10.9 史托克思定理

11.1傅立葉級數
11.2週期為2L之函數
11.3.偶函數與奇函數‧半程展開式
11.6三角多項式之近似
11.7傅立葉積分

Chapter 9 Vector Differential Calculus, Gradient, Div, Curl
9.1 Vectors in 2-space and 3-space
9.2 Inner Product (Dot Product)
9.3 Vector Product (Cross Product)
9.4 Vector and Scalar Functions and Fields. Derivatives
9.5 Curves. Arc Length. Curvature. Torsion
9.6 Calculus Review: Functions of Several Variables
9.7 Gradient of a Scalar Field. Directional Derivative
9.8 Divergence of a Vector Field
9.9 Curl of a Vector Field
Chapter 10 Vector Integral Calculus. Integral Theorem
10.1 Line Integrals
10.2 Path Independence of Line Integrals
10.3 Calculus Review: Double Integrals
10.4 Green’s Theorem in the Plane
10.5 Surface for Surface Integrals
10.6 Surface Integrals
10.7 Triple Integrals. Divergence Theorem of Gauss
10.8 Stokes’s Theorem
Chapter 11 Fourier Series, Integral, and Transform
11.1 Fourier Series
11.2 Function of Any Period p＝2L
11.3 Even and Odd Functions. Half-Range Expansions
11.6 Approximation by Trigonometric Polynomials
11.7 Fourier Integral

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