# Math Calculus

## Math Calculus

MATH2004 Notes - By Eric Hua
Contents
0.1. Fourier Series

3

0.2. Deﬁnition of Fourier Series.

4

0.3. Geometric interpretation of Fourier series.

8

0.4. Fourier cosine and Fourier sine series.

8

0.5. Convergence of Fourier Series.

11

0.6. Termwise diﬀerentiation and integration.

12

10.1. Curves Deﬁned By Parametric Equations

13

10.2. Calculus with Parametric Equations

14

10.3. Polar Coordinates

17

10.4. Areas and Lengths in Polar Coordinates

20

12.1. Three-Dimensional Coordinate Systems

24

12.2. Vectors

24

12.3. The Dot Product

25

12.4. The Cross Product

26

12.5 Equations of Lines and Planes

27

29

13.1 Vector Functions and Space Curves

33

13.2 Derivatives and integrals of vector equations

34

13.3 Arc Length and Curvature

35

1

14.1. Functions of Several Variables

39

14.2. Limits and Continuity

39

14.3. Partial Derivatives

40

14.4. Tangent Plane and Linear Approximation

41

14.5. The Chain Rule

43

14.6. Directional Derivatives and the Gradient Vector

45

14.7. Maximum and Minimum Values

47

14.8. Lagrange Multipliers

50

15.1. Double Integrals over Rectangles

53

15.2. Iterated Integrals

53

15.3. Double Integrals over General Regions

54

15.4. Double Integrals in Polar Coordinates

56

15.7. Triple Integrals
15.7.1 Triple Integral over a Rectangular Box . . . . . . . . . . . . . . . . . . .
15.7.2 Triple Integrals over a General Region . . . . . . . . . . . . . . . . . . . .

57
57
58

15.8. Triple Integrals in Cylindrical Coordinates

60

15.9. Triple Integrals in Spherical Coordinates

62

16.1. Vector Fields

65

16.2. Line Integrals

65

16.3. The Fundamental Theorem for Line Integrals

68

16.4. Green’s Theorem

70

2

0.1. Fourier Series
Pre-knowledge
1. Trig Identities:
π
) = (−1)n ; cos(nπ ) = (−1)n , n...