Richard Wong

Assistant Adjunct Professor in Mathematics at UCLA.

You can find the course syllabus here.

You can view the course lecture notes for 32AH 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?

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 behavior of electromagnetic forces, or the path of steepest ascent 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 32AH differs from 32A in that it covers the topics of multivariable calculus with more mathematical rigor. Moreover, it builds the foundation for more advanced topics, such as linear algebra, real analysis, and differential geometry.

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

Schedule

  Learning Outcome Textbook Section Lectures
MV1 $\mathbb{R}^n$ as a vector space. Geometrically interpret and reason with vector space axioms.  Prove that sets are vector spaces or subspaces.  Determine if a set of vectors are linearly independent, or a basis.  Calculate coordinate vectors.    1-3
MV2 Linear maps and matrices. Determine if a function from $\mathbb{R}^n \to \mathbb{R}^m$ is linear.  Determine the standard matrix associated to a linear map. Add and multiply matrices. Verify that a linear map is invertible. Compute determinants of $2\times2$ and $3\times3$ matrices.   3-5
MV3 Vector operations in $\mathbb{R}^n$. Compute and interpret dot products, the projection of a one vector onto another vector, and the angle between two vectors.  State and use the Cauchy-Schwarz and triangle inequalities. Determine the equations of lines and  planes in space. Compute cross products. Compute the volume of a parallelpiped.   6-9
MV4 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.   10-15
MV5 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.   16-19
MV6 Local optimization. Compute directional derivatives and the gradient.  Interpret level curves of multivariable functions. Find and classify local extrema of a multivariable function.   20-22
MV7 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.    23-25

Learning Objectives:

The goals of the course are that you: 

  • Acquire an understanding of the geometry of Euclidean space, linear transformations, and the differential calculus of multivariable functions.
  • 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. $\mathbb{R}^n$ as a vector space. Geometrically interpret and reason with vector space axioms.  Prove that sets are vector spaces or subspaces.  Determine if a set of vectors are linearly independent, or a basis.  Calculate coordinate vectors. 
  2. Linear maps and matrices. Determine if a function from $\mathbb{R}^n \to \mathbb{R}^m$ is linear.  Determine the standard matrix associated to a linear map. Add and multiply matrices. Verify that a linear map is invertible. Compute determinants of $2\times2$ and $3\times3$ matrices.
  3. Vector operations in $\mathbb{R}^n$. Compute and interpret dot products, the projection of a one vector onto another vector, and the angle between two vectors.  State and use the Cauchy-Schwarz and triangle inequalities. Determine the equations of lines and  planes in space. Compute cross products. Compute the volume of a parallelpiped.
  4. 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.
  5. 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.
  6. Local optimization. Compute directional derivatives and the gradient.  Interpret level curves of multivariable functions. Find and classify local extrema of a multivariable function.
  7. 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. 

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.