Lectures Notes: Topics in Vector Calculus

MA 510, Vector Calculus, Spring 2025.

 Midterm 1: Thursday February 20, 2025. Time: 8:00-9:30 p.m. Location: LWSN B155

 Midterm 2: Thursday April 10, 2025. Time: 8:00-9:30 p.m. Location: LWSN B155

 Office hours: Wednesday 1:30-2:30 p.m. or by appointment

 Class Policy

 

Book: Jerrold E. Marsden and Anthony J. Tromba, Vector Calculus, Sixth Edition, W.H. Freeman and Company, New York.

Section 1.1: Vectors 

Section 1.2: Inner Product

Section 1.3: Matrices, Determinants, and Cross Product

Section 1.4: Cylindrical and Spherical Coordinates

Section 2.1: Real-Valued Functions

Section 2.2: Limits and Continuity

Section 2.3: Differentiation 

Section 2.3: Differentiation, continuation

Section 2.4: Paths and Curves 

Section 2.5: Properties of Derivatives

Section 2.6: Gradients and Directional Derivatives

Section 3.1: Iterated Partial Derivatives

Section 3.2: Taylor's Theorem

Section 3.3: Extrema of Real-Valued Functions

Section 3.4: Constrained Extrema and Lagrange Multipliers

Section 4.1: Acceleration and Newton's Second Law

Section 4.2: Arc Lenght

Section 4.3: Vector Fields 

Section 4.4: Divergence and Curl 

Sections 5.1 and 5.2: Double Integral over a Rectangle 

Sections 5.3 and 5.4: Doble Integral Over More General Regions, Changing the Order of Integration

Section 5.5: Triple Integral

Section 6.1: Geometry of Maps

Section 6.2: The Change of Variables Theorem

Sections 6.2: The Change of Variable Theorem, continuacion

Sections 7.1 and 7.2: Path Integral, Line integral

Section 7.2: Line integral, continuation

Section 7.3: Parametrized Surfaces

Section 7.4: Area of a Surface

Section 7.5: Integrals of Scalar Functions over Surfaces

Section 7.6: Surface Integrals of Vector Fields

Section 8.2: Stokes'Theorem

Section 8.3: Conservative Fields

Section 8.1: Green's Theorem

Section 8.4: Gauss' Theorem

Remarks on the Divergence Theorem

Applications of the Divergence Theorem and Stokes' Theorem

Homework

HW 1: Due on Friday January 24

1.1: 13, 17, 21, 25

1.2: 11, 13, 15, 17, 21

1.3: 6, 7, 16a, 17, 29, 30, 35

1.4: 1, 8

1.5: 3, 7

HW 2: Due on Friday January 31

2.1: 3, 6, 8, 9, 13, 21, 25, 33, 42

2.2: 1, 2, 6, 9(a), 25, 33.

2.3: 2, 5, 9, 12, 13

HW 3: Due on Friday February 7

2.3: 16, 17, 22, 25, 27

2.4: 1, 3, 5, 7, 13, 16, 17, 19, 23

2.5: 2(f), 3(b), 8, 10, 13, 15, 17, 19, 20, 22, 26

HW 4: Due on Friday February 14

2.6: 3, 4, 5, 7, 8(a), 15, 22(a), 31

3.1: 4, 11, 12, 22, 25, 31, 32

3.2: 2, 3, 8

HW 5: Due on Monday February 24

3.3: 13, 23, 31, 41

3.4: 2, 3, 13

HW 6: Due on Saturday March 1

4.1: 1, 4, 6, 9, 10

4.2: 3, 9, 13, 17

4.3: 9, 10, 18, 22

4.4: 1, 4, 14, 15, 22, 25, 27, 28

HW 7: Due on Friday March 7

5.1: 1(b), 14

5.2: 1(a), 1(b), 4

HW 8: Due on Sunday March 23

5.3: 1, 2, 4(a)(c)(e), 6, 15

5.4: 1, 2, 3

5.5: 1, 6, 8, 9, 13, 17, 18, 24

6.1: 3, 5, 6, 7

6.2: 3, 4, 7, 25, 26, 30, 33

HW 9: Due on Friday March 28

7.1: 10, 18(a), 20

7.2: 1, 3, 5, 11, 18, 19

HW 10: Due Friday April 4

7.3: 3, 7, 9, 14, 15, 18, 22(a)

7.4: 1, 5, 7, 13

7.5: 3, 5, 7, 16

7.6: 5, 7, 8, 13