Math 32A: Calculus of Several Variables

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 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.

Schedule

	Learning Outcome	Textbook Section	Lectures
MV1	Vector operations. Compute and interpret dot products, cross products, the projection of a one vector onto another vector, and volume.	13.1-13.4	1-5
MV2	Planes and surfaces. Determine the equations of lines and planes in space; recognize the standard quadric surfaces. Draw graphs of planes, cylinders, and quadric surfaces.	13.5-13.6	6-7
MV3	Calculus of vector-valued functions. Compute and interpret derivatives of vector-valued functions; Compute tangent vectors to parametric curves; determine velocity, speed, and acceleration.	14.1-14.3	8-9
MV4	Arclength and curvature. Compute the arclength of a curve. Compute and geometrically interpret the unit tangent, unit normal vector, and curvature for a curve.	14.3-14.5	10-13
MV5	Multivariable functions. Draw level curves and surfaces of multivariable functions. Compute limits of multivariable functions; determine if multivariable functions are continuous.	15.1-15.2	13-16
MV6	Partial derivatives and directional derivatives. Compute and geometrically interpret partial derivatives, directional derivatives, and the gradient. Compute derivatives using various chain rules.	15.3, 15.5-15.6	16-18
MV7	Local optimization. Compute and geometrically interpret the gradient. Interpret level curves of multivariable functions. Find and classify local extrema of a multivariable function.	15.5, 15.7	19-21
MV8	Constrained optimization. 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.	15.8	22-25
MV9	Linear approximation and the multivariable derivative. Find equations for tangent planes to surfaces at a given point. Compute linear approximations of a multivariable function at a given point. Determine if a multivariable function is differentiable. Compute and use the Jacobian matrix.	15.4	25-28

Learning Objectives:

The goals of the course are that you:

Acquire an understanding of the geometry of space, vectors, and the differential calculus of vector functions and 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)

Vector operations. Compute and interpret dot products, cross products, the projection of a one vector onto another vector, and volume.
Planes and surfaces. Determine the equations of lines and planes in space; recognize the standard quadric surfaces. Draw graphs of planes, cylinders, and quadric surfaces.
Calculus of vector-valued functions. Compute and interpret derivatives of vector-valued functions; Compute tangent vectors to parametric curves; determine velocity, speed, and acceleration.
Arclength and curvature. Compute the arclength of a curve. Compute and geometrically interpret the unit tangent, unit normal vector, and curvature for a curve.
Multivariable functions. Draw level curves and surfaces of multivariable functions. Compute limits of multivariable functions; determine if multivariable functions are continuous.
Partial derivatives and directional derivatives. Compute and geometrically interpret partial derivatives, directional derivatives, and the gradient. Compute derivatives using various chain rules.
Local optimization. Compute and geometrically interpret the gradient. Interpret level curves of multivariable functions. Find and classify local extrema of a multivariable function.
Constrained optimization. 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.
Linear approximation and the multivariable derivative. Find equations for tangent planes to surfaces at a given point. Compute linear approximations of a multivariable function at a given point. Determine if a multivariable function is differentiable. Compute and use the Jacobian matrix.

Mathematical Reasoning Objectives (MR)

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.
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.
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.
Use appropriate tools strategically. Consider the available tools when solving a mathematical problem. Understand and use stated assumptions, definitions, and previously established results.
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.
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)

Ask questions. Notice, identify, and clarify sources of confusion. Look for connections and relationships between ideas. Explore topics and ideas deeply.
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.
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.
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.
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.
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.
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.

Richard Wong