MATH 158 - Calculus III

Gauss

Stokes

Suggested Textbook

I will refer in my lectures to the following freely available textbook:

• J. Marsden & A. Weinstein, Calculus III, Caltech
• J. Marsden & A. Weinstein, Calculus III Students' Guide, Caltech

Since Calculus III did not change in the last century, you can study instead from the Stewart's textbook if you like to. In fact, I do suggest you all to study from both of them since being exposed to a different point of view is always very good! If you like I list below a series of alternate free textbooks you are also always welcome to consult.

Syllabus

Get here a copy of the full syllabus. See below the lectures plan:

Week	Content	Topics
1	13.1,13.2,13.3	Vectors in plane and space, lines and distance.
2	13.4,13.5,13.6	Scalar and vector products, matrices and determinants.
3	14.3,14.4,14.5	Functions in two variables, level sets, quadrics, cylindrical and spherical coordinates.
4	14.6, 14.7,15.1	Curves in space and their arc length. Partial Derivatives.
5	15.2, 15.3,15.4	Tangent plane, chain rule and matrix multiplication. Midterm 1.
6	16.1, 16.2	Gradients and directional derivatives, level surfaces and implicit differentiation.
7	16.3, 16.4	Maxima and minima, Lagrange multipliers.
8	17.1,17.2	Double integrals. Midterm 2.
9	Spring Break
10	17.3, 17.4	Applications of double integrals. Triple integrals.
12	17.5, 17.6	Integrals in polar, cylindrical and spherical coords. Applications of triple integrals.
12	18.1, 18.2	Line integrals. Path independence.
13	18.3, 18,4	Exact Differentials, Green's Theorem.
14	18.5, 18,6	Circulation and Flux of a vector field.
15	Review	Midterm 3.
16	Review, Reading

Other free texts in pdf

• Paul Dawkings, Calculus III, Lamar University
• Jerry Shurman, Multivariable Calculus, Reed College
• Sergey Shabanov, Concepts in Calculus III, U. of Florida
• Michael Corral, Vector Calculus, Schoolcraft College
• Dan Sloughter, The Calculus of Functions of Several Variables, Furman University
• David Guichard, Single and Multivariable Calculus, Whitman College
• Zbigniew Nitecki, Calculus in 3D, Tufts University
• George Cain & James Herod, Multivariable Calculus, Georgia Tech

Homework

Graded Homework will be assigned weekly on Webwork.

I will usually provide you a short list of ungraded homework from the suggested textbook. While Webwork problems will usually be straightforward problems, these ungraded ones will be usually harder and are more meant to make you think more about and understand more the topics of the corresponding sections. The list of those problems is in the lectures log below.

Video lectures

Here are a few series of very well done Video Lectures you might use in case you feel the need of hearing more about any of the topics we are going to cover:

• Calculus 3 Lecture Videos from University of Utah
• Calc III Video Lectures from MIT
• Michael Hutchings, Multivariable Calculus Video Lectures, University of California, Berkeley

Lectures log

), Line Integrals

Week	Day	Sections	Topics	Exercises
1	13 Jan	13.1	Vectors in 2D. Sum of two vectors. Product of a vector by a scalar.	13.1: 27,28,29,37
	15 Jan	13.2, 13.3	Vectors in 3D. Sum of two vectors. Product of a vector by a scalar. Parametric equation of a line	13.2: 23, 42
	16 Jan	13.3, 13.4	Dot product of two vectors. Geometric meaning of the dot product. Length of a vector in 2D and 3D. Equation of a plane perpendicular to a 3D vector.	13.3: 25,30 13.4: 35,47,48
	17 Jan	13.5, 13.6	Relative position of lines and planes in 3D space.
2	22 Jan	13.4	Cross product. Writing the components of the cross product through the determinant of 2x2 and 3x3 matrices..	13.5: 20,34
	23 Jan	14.3	Triple product and its geometrical meaning. Functions in two variables. Graphs and level curves of functions in two variables.	14.3: 24, 31
	24 Jan	14.4	Graph and level sets of linear and quadratic functions, Quiz 1	14.4: 23, 24, 25
3	27 Jan	14.4,14.5	Quadrics: Ellipsoid, 1- and 2-sheeted Hyperboloids. Polar coordinates	14.5: 39, 40, 59
	29 Jan	14.5	Elliptic and Hyperbolic paraboloid, cilinders. Polar coordinates.	14.6: 1, 3
	30 Jan	14.5	Cilindrical and Spherical coordinates	14.6: 22, 33, 43
	31 Jan Feb	14.6	Parametric equations of curves, Quiz 2.
4	3 Feb	14.7	Length of a curve.	14, p. 763: 73, 74
	5 Feb	14.7	Arc length.
	6 Feb	14.7	Arc length
	7 Feb	15.1	Partial Derivatives	15.1: 75, 76, 77
5	10 Feb	15.2	Tangent planes, Linearization, Taylor series	15.2: 27, 29
	12 Feb	15.3	The chain rule, Quiz 3
	13 Feb	15.4	Matrix multiplication and the chain rule
	14 Feb		President Day, no class
6	19 Feb	16.1	Gradient, antiderivatices
	20 Feb	16.2	Differentiability, Quiz 4
	21 Feb		Review of Chapters 13, 14 & 15 -- Practice Exam, solutions
7	24 Feb		Midterm 1
	26 Feb	16.2	Implicit differentiation	16.2: 50, 51
	27 Feb	16.3	Maxima, minima and saddle points	16.3: 51, 53
	28 Feb	16.4	Maxima, minima and saddle points, Quiz 5
8	2 Mar	16.4	Constrained extrema and Lagrange multipliers	16.4: 23
	4 Mar	16.4	Constrained extrema and Lagrange multipliers, Quiz 6
	5 Mar	15.1	Continuity, Limits, Chain rule
	6 Mar	15.1	Continuity, Limits
11	9 Mar	15.1	Continuity, Limits
	11 Mar		Review of Chapters 15 & 16 -- Practice Exam and its solutions
	12 Mar		Midterm 2
	13 Mar	17.1	Double integrals
9	14--22 Mar		Spring Break
11	23 Mar	17.2	Double integral over a general region
	25 Mar	17.2	Double integral over a general region
	26 Mar	17.3	Applications of double integrals
	27 Mar	17.5	Double integrals in Polar coordinates
12	30 Mar		Solutions of Webwork problems, Quiz 7
	1 Apr		Discussion of problems in Quiz 7
	2 Apr	17.4	Triple integrals
	3 Apr	17.5	Applications of triple integrals
12	6 Apr	17.5	Changes of variables in triple integrals, Quiz 8
	8 Apr	18.1	General change of variables in double integrals (study here
	9 Apr	18.2	Conservative Vector Fields
	10 Apr	18.3,18.4	Exact Differentials, Green's Theorem
13	13 Apr	18.4	Green's Theorem, Quiz 9
	15 Apr	18.5	Circulation and Stoke's Theorem
	16 Apr	18.5,18.6	Circulation and Stoke's Theorem, Flux and Divergence Theorem
	17 Apr	18.6	Flux and Divergence Theorem
14	20 Apr		Review, Quiz 10
	22 Apr		Review of Chapters 17 and 18 -- Practice Exam, solutions
	23 Apr		Midterm 3
	24 Apr		Review of all chapters