Richard Wong

Assistant Adjunct Professor in Mathematics at UCLA.

You can find the course syllabus here.

Course Description:

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.

Schedule

  Learning Outcome Textbook Section Lectures
MV1 Integration over regions in $\mathbb{R}^2$ and $\mathbb{R}^3$. Set up and evaluate integrals of multivariable functions over regions in the plane (double integrals) and over regions in space (triple integrals) using iterated integrals. 16.1-16.2 1-5
MV2 Integration in polar, cylindrical, and spherical coordinates. Use the polar coordinate system to set up and evaluate double integrals. Use cylindrical and spherical coordinate systems to set up and evaluate triple integrals. 12.3, 13.7, 16.4 6-7
MV3 Change of variables. Set up and use the change of variables formula to evaluate double and triple integrals. Compute and geometrically interpret the Jacobian. 16.6 8-9
MV4 Vector fields. 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. 17.1 10-13
MV5 Line integrals. Calculate line integrals of scalar functions. Calculate line integrals of vector fields.  Use the line integral to calculate work. Given a conservative vector field, find an associated potential function. Use the fundamental theorem of line integrals to evaluate the line integral of a conservative vector field. 17.2-17.3 13-16
MV6 Parametrized surfaces and surface integrals. Use vector parametrization of a surface to find the tangent plane to the surface at a point.  Use vector parametrization to compute surface area and other surface integrals. 17.4-17.5 17-19
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. 18.1-18.3 20-26

Learning Objectives:

The goals of the course are that you: 

  • Acquire an understanding of the geometry of space and the integral calculus of multivariable functions and vector fields.
  • 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 $\mathbb{R}^2$ and $\mathbb{R}^3$. Set up and evaluate integrals of multivariable functions over regions in the plane (double integrals) and over regions in space (triple integrals) using iterated integrals.
  2. Integration in polar, cylindrical, and spherical coordinates. Use the polar coordinate system to set up and evaluate double integrals. Use cylindrical and spherical coordinate systems to set up and evaluate triple integrals.
  3. Change of variables. Set up and use the change of variables formula to evaluate double and triple integrals. Compute and geometrically interpret the Jacobian.
  4. Vector fields. 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.
  5. Line integrals. Calculate line integrals of scalar functions. Calculate line integrals of vector fields.  Use the line integral to calculate work. Given a conservative vector field, find an associated potential function. Use the fundamental theorem of line integrals to evaluate the line integral of a conservative vector field.
  6. Parametrized surfaces and surface integrals. Use vector parametrization of a surface to find the tangent plane to the surface at a point.  Use vector parametrization to compute surface area and other surface integrals.
  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.