Multivariable Calculus

Students enhance their understanding of vector-valued functions to include analyzing limits and continuity with vector-valued functions, applying rules of differentiation and integration, unit tangent, principal normal and binormal vectors, osculating planes, parametrization by arc length, and curvature.

As part of this course, students undertake a deep dive into the differential calculus of multivariable functions, including partial derivatives, tangent planes, linear approximations, the gradient vector, directional derivatives, the multivariable chain rule, and differentials. Students then generalize their understanding of derivatives of multivariable and vector-valued functions to include the total derivative of a function, second derivatives, and multivariable Taylor polynomials.

Students develop their knowledge of optimization techniques to include unconstrained optimization of multivariable functions and the role of Taylor polynomials and the Hessian determinant in classifying critical points. Students will also solve constrained optimization problems using the method of Lagrange multipliers.

This course extends students' understanding of integration to multiple integrals, including their formal construction using Riemann sums, calculating multiple integrals over various domains, and applications of multiple integrals. Students explore affine and nonlinear transformations in the plane and use this to understand the role of the Jacobian in performing changes of variables with multiple integrals, including cylindrical and spherical polar coordinates.

The course concludes with an exploration of vector fields, line integrals for scalar and vector-valued functions, the fundamental theorem of line integrals, circulation and flux in two-dimensional and three-dimensional vector fields, Green's theorem, surface integrals, Stokes' theorem, and the divergence theorem.

• Extend previous understandings of vector-valued functions to include functions with three spatial components and compute their domains, limits, derivatives, and integrals, and describe conditions for continuity and differentiability of these functions.
• Analyze space curves, including understanding and computing unit tangent, principal normal, and binormal vectors, osculating planes, and parameterizing a curve by arc length.
• Understand curvature and radius of curvature, and compute these quantities for various curves.
• Learn essential concepts related to multivariable functions, including their various representations, domains, ranges, limits, and continuity.
• Investigate important quadric surfaces and learn how to compute properties such as intersections, traces, and projections of these surfaces onto various planes.
• Compute partial derivatives, interpret their meaning geometrically, and apply Clairaut's theorem.
• Find tangent planes to multivariable functions, and understand how these relate to linear approximations.
• Understand the role of the gradient vector, calculate gradient vectors, and interpret them appropriately. Understand how the gradient vector relates to the directional derivative.
• Use the gradient vector to compute tangent and normal lines and planes to level curves and surfaces.
• Apply the multivariable chain rule, including its extension to polar coordinates, implicit differentiation, and differentials.
• Compute derivatives of multivariable and vector-valued functions, understand their role in forming linear approximations, and their relationship with the gradient vector.
• Compute second derivatives of multivariable functions.
• Form quadratic Taylor polynomials of multivariable functions in scalar and vector form, and use these to form quadratic approximations of functions.
• Understand the definitions of global and local extrema of multivariable functions.
• Classify the critical points of unconstrained multivariable functions using their Taylor polynomials and the second partial derivatives test. Understand how the second partial derivatives test relates to quadratic forms.
• Master the method of Lagrange multipliers for one and multiple constraints.
• Describe the effects of affine and nonlinear transformations on objects in the plane.
• Calculate the Jacobian determinant of a transformation and use it to find critical points.
• Approximate volumes using Riemann sums and construct double integrals using lower and upper bounds.
• Calculate double integrals over rectangular domains, Type I, and Type II regions, and change the order of integration.
• Apply the change of variables formula for double integrals, including converting to plane polar coordinates.
• Calculate triple integrals over rectangular domains, Type I, Type II, and Type III regions, and change the order of integration.
• Master change of variables for triple integrals, including converting to cylindrical polar and spherical polar coordinates.
• Calculate the divergence and curl of a vector field, and describe qualitatively and quantitatively the effects these quantities describe.
• Understand gradient, conservative, and source-free vector fields, and calculate potential and stream functions.
• Compute line integrals of scalar functions in various forms, and understand their properties.
• Compute line integrals of vector-valued functions and understand how these can be used to find circulation and flux in two-dimensional vector fields.
• Apply the fundamental theorem of line integrals, and understand path independence.
• Master Green's theorem conceptually and computationally, including Green's theorem in flux form and methods of extension.
• Understand parametric surfaces, and parametrize quadric and other surfaces.
• Compute surface areas of surfaces and surface integrals of scalar-valued functions.
• Compute the flux of a vector field through a surface.
• Apply Stokes' theorem and relate it to Green's theorem in circulation form.
• Understand and apply the divergence theorem, and relate it to Green's theorem in flux form.
• Apply multivariable calculus concepts to real-world problems in physics, engineering, statistics, and other fields. These include calculating quantities such as mass and charge using density functions, centers of mass, moments of inertia, radii of gyration, finding the average value of a function, and computing probabilities using joint probability density functions.