Richard Wong

Assistant Adjunct Professor in Mathematics at UCLA.

You can find the course syllabus here.

You can view the course lecture notes for 32BH here.

How can we describe the physical world mathematically? What changes, and what stays the same when we move from single variable calculus to multivariable calculus? What does it mean to take a integral of a multivariable function? What kinds of functions can we integrate? How far can we generalize the notion of integration?

Multivariable calculus is the mathematical language that allows us to describe the geometry of the physical world around us, such as the areas, volumes, or mass of objects; the behaviors of electromagnetic fields or fluids in space; or calculating the amount of wind blowing through the hills of Los Angeles. In this course, you will develop the reasoning and questioning skills needed to explore these geometric concepts and apply them to real-life situations. Moreover, you will become fluent in communicating your ideas through the mathematical language of multivariable calculus.

The course 32BH differs from 32B in that it covers the topics of multivariable calculus with more mathematical rigor. In particular, we will focus more on learning how to grapple with and understand complex mathematical concepts, as well as how to explore and generalize theorems of integration. Moreover, this course builds the foundation for more advanced topics, such as real analysis, complex analysis, and differential geometry.

This course is recommended for students interested in learning about advanced mathematics.

Schedule

  Learning Outcome Textbook Section Lectures
MV1 Integration over regions in $\mathbb{R}^n$. Use Darboux sums to define and estimate integrals. Set up and evaluate integrals of continuous multivariable functions over regions in $\mathbb{R}^n$ using iterated integrals.  Use Fubini’s theorem to evaluate integrals.  Use integrals in applications to physics and real-world scenarios.    1-4
MV2 Integrability. Explore the definition of the multivariable integral.  Determine whether or not a function is integrable.  Generate examples of integrable, non-continuous functions.   5-6
MV3 Change of variables. Use the polar, cylindrical, and spherical coordinate systems to set up and evaluate iterated integrals. Set up and use the change of variables formula to evaluate double and triple integrals. Compute and geometrically interpret the Jacobian.   7-9
MV4 Arclength and Surface Integrals. Parametrize curves in $\mathbb{R}^n$ and surfaces in $\mathbb{R}^3$.  Calculate line integrals of scalar functions (including arclength). Use the parametrization of a surface to find the tangent plane to the surface.  Compute surface integrals of scalar functions (including surface area).   10-12
MV5 Manifolds in $\mathbb{R}^n$. Understand how the definition of manifold in $\mathbb{R}^n$ generalizes curves and surfaces. Determine if a subset of $\mathbb{R}^n$ is a manifold.  Use the parametrization of a manifold to integrate scalar functions over manifolds.   13-18
MV6 Vector fields; Flow and Flux integrals. Sketch and visualize vector fields.  Evaluate and interpret the physical meaning of the curl and divergence of a vector field. Determine whether or not a vector field is conservative. Given a conservative vector field, find an associated potential function. Calculate line integrals of vector fields. Use the fundamental theorem of line integrals to evaluate the line integral of a conservative vector field. Calculate flux integrals of vector fields over curves and surfaces.   19-22
MV7 Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. Geometrically interpret Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem.  Identify the appropriate theorem to use in order compute a particular line integral or surface integral of a vector field.   23-25

Learning Objectives:

The goals of the course are that you: 

  • You acquire an understanding of the integral calculus of multivariable functions in $\mathbb{R}^n$; the geometry of curves, surfaces, and manifolds; and the notion of integration on manifolds;
  • Develop the reasoning and questioning skills needed to explore these (mathematical) topics and apply them to real-life situations.
  • Develop the collaboration and communication skills needed to convey your (mathematical) ideas.

Below you will find the explicit learning objectives associated to each of these goals.

Multivariable Calculus Objectives (MV)

  1. Integration over regions in $\mathbb{R}^n$. Use Darboux sums to define and estimate integrals. Set up and evaluate integrals of continuous multivariable functions over regions in $\mathbb{R}^n$ using iterated integrals.  Use Fubini's theorem to evaluate integrals.  Use integrals in applications to physics and real-world scenarios. 
  2. Integrability. Explore the definition of the multivariable integral.  Determine whether or not a function is integrable.  Generate examples of integrable, non-continuous functions.
  3. Change of variables. Use the polar, cylindrical, and spherical coordinate systems to set up and evaluate iterated integrals. Set up and use the change of variables formula to evaluate double and triple integrals. Compute and geometrically interpret the Jacobian.
  4. Arclength and Surface Integrals. Parametrize curves in $\mathbb{R}^n$ and surfaces in $\mathbb{R}^3$.  Calculate line integrals of scalar functions (including arclength). Use the parametrization of a surface to find the tangent plane to the surface.  Compute surface integrals of scalar functions (including surface area).
  5. Manifolds in $\mathbb{R}^n$. Understand how the definition of manifold in $\mathbb{R}^n$ generalizes curves and surfaces. Determine if a subset of $\mathbb{R}^n$ is a manifold.  Use the parametrization of a manifold to integrate scalar functions over manifolds.
  6. Vector fields; Flow and Flux integrals. Sketch and visualize vector fields.  Evaluate and interpret the physical meaning of the curl and divergence of a vector field. Determine whether or not a vector field is conservative. Given a conservative vector field, find an associated potential function. Calculate line integrals of vector fields. Use the fundamental theorem of line integrals to evaluate the line integral of a conservative vector field. Calculate flux integrals of vector fields over curves and surfaces.
  7. Green's Theorem, Stokes' Theorem, and the Divergence Theorem. Geometrically interpret Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem.  Identify the appropriate theorem to use in order compute a particular line integral or surface integral of a vector field.

Mathematical Reasoning Objectives (MR)

  1. Reason abstractly and quantitatively.  Use mathematics to model real-world situations and to interpret and solve problems.  Make appropriate assumptions and approximations to simplify a complicated situation. Draw pictures or study examples to provide insight.  Attend to the meaning of quantities instead of just computing them. Consider the units involved.
  2. Make sense of problems and persevere in solving them.  Understand what a problem is asking for. Analyze the givens, constraints, and goals of a problem. Break large/complex problems into smaller/simpler problems. Check answers using alternate methods.
  3. Build intuition.  Investigate and create specific examples and counterexamples. Look for and make use of structure or repeated patterns. Look for both general methods and shortcuts.  Seek to understand unexpected results.
  4. Use appropriate tools strategically.  Consider the available tools when solving a mathematical problem. Understand and use stated assumptions, definitions, and previously established results.
  5. Construct viable arguments.  Make conjectures, and use assumptions, definitions, and/or previously established results to prove or disprove them. Use strategies such as direct proofs, contrapositive statements, proof by cases, or proof by contradiction.
  6. Be creative and explore mathematics. Do more than find a solution. Look for novel or elegant solutions. See what changes when you add, change, or weaken hypotheses. Draw connections between topics and ideas.

Mathematical Communication Objectives (MC)

  1. Ask questions.  Notice, identify, and clarify sources of confusion. Look for connections and relationships between ideas. Explore topics and ideas deeply.
  2. Use and develop mathematical fluency.  Use standard mathematical notations and terms (as discussed in class or demonstrated in course materials).  Clearly indicate and explain any use of non-standard shorthand, notation, or tools.
  3. Analyze and constructively critique the reasoning of others.  Actively listen and summarize key ideas to check comprehension. Test conjectures against examples and potential counterexamples. Assess and reconcile various approaches to problems. Work together to find errors and fix flaws.
  4. Explain and justify your reasoning.  Communicate and justify your conclusions to others. Indicate the general strategy or argument, and identify the key step or idea(s). Listen and reflect to the critiques of others.  Work together to find errors and fix flaws.
  5. Attend to precision.  Communicate precisely to others. Use clear definitions, and carefully state any assumptions or results used.  Be able to explain heuristics/arguments in depth.
  6. Be clear and concise.  se the appropriate amount of generality or specificity in arguments.  Avoid use of any extraneous assumptions, hypotheses, or statements. Indicate any figures or examples that you have in mind.
  7. Review, reflect, and revise. Review previous work and assess the positives and negatives. Reflect on your strategies and look for ways to improve. Use feedback to grow and develop your mathematical reasoning or communication skills.