Course Description

Vector Calculus, MATH 23B

Syllabus(opens in a new tab) Notify me when available

Double integral, changing the order of integration. Triple integrals, maps of the plane, change of variables theorem, improper double integrals. Path integrals, line integrals, parametrized surfaces, area of a surface, surface integrals. Green's theorem, Stokes' theorem, conservative fields, Gauss' theorem. Applications to physics and differential equations, differential forms.

Key Information

Academic Calendar(opens in a new tab)

Credit: 5 quarter units / 3.33 semester units credit
UC Santa Cruz, Mathematics

Course Credit:

Upon successful completion, all online courses offered through cross-enrollment provide UC unit credit. Some courses are approved for GE, major preparation and/or, major credit or can be used as a substitute for a course at your campus.

If "unit credit" is listed by your campus, consult your department, academic adviser or Student Affairs division to inquire about the petition process for more than unit credit for the course.

UC Berkeley:
Unit Credit

UC Davis:
Course Equivalence: UCD MAT 021D
General Education: QL, SE.

UC Irvine:
Course Equivalence: UCI - MATH 2E

UC Los Angeles:
Course Equivalence: Mathematics 32B
Quantitative Reasoning

UC Merced:
Course Equivalence: UCSC MATH 23A+23B = UCM MATH 023
Units toward degree (see your advisor)

UC Riverside:
Course Equivalence: MATH 10B

UC San Diego:
Course Equivalence: UCSD Math 20E
General Education: Revelle - completes Math GE; Warren - May be used depending on major/PofC; TMC - Clears one TMC MATH/LOGIC GE requirement from the Math, Advanced Statistics area; ERC - formal skills, Seventh - 1 course towards Alternatives - Quantitative Reasoning; Muir: 1 course in a Natural Sciences theme in "Math and Statistics"

UC San Francisco:
Unit Credit

UC Santa Barbara:
Course Equivalence: Math 6B at UCSB
General Education: This course will apply to Area C and Quantitative Relationships automatically upon completion

UC Santa Cruz:
General Education: MF
Major Requirement: Lower-division Problem Solving requirement for Mathematics Majors

Course Fees

There is a fee of about $80 for the customized interactive E-text which includes access to the online homework, reading assignments as well as the online quizzes and exams.

More About The Course

With a focus on the integral calculus of functions of several variables, in Math 23B online you will learn several generalisations of the study of the integral of a single variable from Calculus 2: 

  • the Double and Triple Integrals
  • Line, Path and Surface Integrals
  • The Fundamental Theorems of Calculus in Several Variables and their Applications to Physics.
The online format allows us to study the historical foundations of the subject and its applications to the physical sciences. The course gives students a greater ability to self-pace their learning, experiment and use technology in order to further their knowledge and understanding through an interactive and dynamic E-Book. The course offers students an online discussion forum where they can post questions relating to the video lectures, homework, reading, and course logistics. Students are encouraged to respond to each others questions, and instructors and TA’s monitor these forums, responding to student questions as well. In addition, the teaching staff hold regular online office hours, as well as optional discussion sections.

Course Creators

Anthony Tromba

Anthony (Tony) Tromba was born and raised in New York where he attended The Brooklyn Technical High School. He completed his undergraduate work in Mathematics at Cornell University and received his Ph.D. in Mathematics from Princeton. His first academic position was as Assistant Professor of Mathematics at Stanford University. He later held the Chair of Analysis at the Ludwig Maximilians Universität in Munich, Germany, and is now a Distinguished Professor of Mathematics at the University of California at Santa Cruz.


He has been a Max Planck research group leader, a member of the technical staff at Bell Labs and the Director of Development of the Mathematical Sciences Research Institute in Berkeley, California. Tony has held visiting professorships at many universities throughout the world, including Universities in Paris, Florence, Moscow, Tokyo, Rio de Janeiro, Beijing, Warsaw, and London, He is the author of nine books including the first Mathematics book in the Scientific American Library series. His Vector Calculus textbook, which appears in six editions and five languages, is used by many of America's leading universities.

 Anthony (Tony) Tromba was born and raised in New York where he attended The Brooklyn Technical High School. He completed his undergraduate work in Mathematics at Cornell University and received his Ph.D. in Mathematics from Princeton. His first academic position was as Assistant Professor of Mathematics at Stanford University. He later held the Chair of Analysis at the Ludwig Maximilians ...

Anthony (Tony) Tromba was born and raised in New York where he attended The Brooklyn Technical High School. He completed his undergraduate work in Mathematics at Cornell University and received his Ph.D. in Mathematics from Princeton. His first academic position was as Assistant Professor of Mathematics at Stanford University. He later held the Chair of Analysis at the Ludwig Maximilians Universität in Munich, Germany, and is now a Distinguished Professor of Mathematics at the University of California at Santa Cruz.


He has been a Max Planck research group leader, a member of the technical staff at Bell Labs and the Director of Development of the Mathematical Sciences Research Institute in Berkeley, California. Tony has held visiting professorships at many universities throughout the world, including Universities in Paris, Florence, Moscow, Tokyo, Rio de Janeiro, Beijing, Warsaw, and London, He is the author of nine books including the first Mathematics book in the Scientific American Library series. His Vector Calculus textbook, which appears in six editions and five languages, is used by many of America's leading universities.


Frank Bauerle

Frank Bäuerle was born and raised in southern Germany. He grew up in Weinsberg, a small town amid castle ruins from the Middle Ages and vineyards that were first cultivated by the Romans when they occupied this land some two thousand years ago. He did his undergraduate work in Mathematics and Computer Science at the Technische Hochschule in Karlsruhe, Germany, after which he received his Ph.D. in Mathematics from The University of California at San Diego.  Frank did his research work in Recursion Theory and Complexity Theory, an area lying at the intersection of Applied Logic and Theoretical Computer Science.

 Frank Bäuerle was born and raised in southern Germany. He grew up in Weinsberg, a small town amid castle ruins from the Middle Ages and vineyards that were first cultivated by the Romans when they occupied this land some two thousand years ago. He did his undergraduate work in Mathematics and Computer Science at the Technische Hochschule in Karlsruhe, Germany, after which he received his ...

Frank Bäuerle was born and raised in southern Germany. He grew up in Weinsberg, a small town amid castle ruins from the Middle Ages and vineyards that were first cultivated by the Romans when they occupied this land some two thousand years ago. He did his undergraduate work in Mathematics and Computer Science at the Technische Hochschule in Karlsruhe, Germany, after which he received his Ph.D. in Mathematics from The University of California at San Diego.  Frank did his research work in Recursion Theory and Complexity Theory, an area lying at the intersection of Applied Logic and Theoretical Computer Science.


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