Richard Wong

Assistant Teaching Professor in Mathematics at Rice University.

You can find the course syllabus 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 derivative of a multivariable function? What does it mean to integrate a multivariable function? What regions can we integrate over? How can we generalize the Fundamental Theorem of Calculus?

Multivariable calculus is the mathematical language that allows us to describe the geometry of the physical world around us, such as the motion of planets in orbit, the path of steepest ascent through the hills of Los Angeles, or how to calculate the amount of electricity flowing along a curve or through a surface. 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 MATH 232 differs from MATH 212 in that it covers the topics of multivariable calculus with more mathematical rigor (e.g. proofs). That is, in addition to learning how to perform standard computations, we will also learn how to grapple with and prove statements about complex mathematical concepts. Moreover, this course builds the foundation for more advanced topics, such as real analysis, complex analysis, and differential geometry. This course differs from MATH 222 in that we will not discuss general manifolds (only curves and surfaces), nor differential forms.

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

Schedule

  Learning Outcome Lectures
MV1 Analysis in $\mathbb{R}^n$. Sketch and geometrically interpret vector-valued and multivariable functions. Compute limits of sequences, limits of multivariable functions; determine if multivariable functions are continuous. 1-8
MV2 The multivariable derivative. Geometrically interpret the derivative of a multivariable function. Compute and use partial derivatives and the Jacobian matrix. Find equations for tangent planes to surfaces and linear approximations of functions at a given point. State and use the chain rule for derivatives. 9-12
MV3 Local Optimization. Compute directional derivatives and the gradient. Interpret level curves of multivariable functions. Find and classify local extrema of a multivariable function. 13-14
MV4 Constrained optimization. Determine if subsets of $\mathbb{R}^n$ are open, closed, bounded, or compact. Use the method of Lagrange multipliers to find local minima and local maxima of functions subject to constraints. Find and classify global extrema on compact domains. Compute and use the Hessian matrix. 15-16
MV5 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. Determine whether or not a function is integrable. Generate examples of integrable, non-continuous functions. 17-21
MV6 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. 22-25
MV7 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). 26-30
MV8 Vector fields; Flow and Flux integrals. Sketch and visualize vector fields. 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. 31-35
MV9 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. 36-40

Learning Objectives:

The goals of the course are that you: 

  • learn how to use the tools of calculus (differentiation and integration) to describe and model the real world.
  • 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. Analysis in $\mathbb{R}^n$. Sketch and geometrically interpret vector-valued and multivariable functions. Compute limits of sequences, limits of multivariable functions; determine if multivariable functions are continuous.
  2. The multivariable derivative. Geometrically interpret the derivative of a multivariable function. Compute and use partial derivatives and the Jacobian matrix. Find equations for tangent planes to surfaces and linear approximations of functions at a given point. State and use the chain rule for derivatives.
  3. Local Optimization. Compute directional derivatives and the gradient. Interpret level curves of multivariable functions. Find and classify local extrema of a multivariable function.
  4. Constrained optimization. Determine if subsets of $\mathbb{R}^n$ are open, closed, bounded, or compact. Use the method of Lagrange multipliers to find local minima and local maxima of functions subject to constraints. Find and classify global extrema on compact domains. Compute and use the Hessian matrix.
  5. 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. Determine whether or not a function is integrable. Generate examples of integrable, non-continuous functions.
  6. 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. 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).
  8. Vector fields; Flow and Flux integrals. Sketch and visualize vector fields. 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.
  9. 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.  Use 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.