Hunter Lehmann

Hunter Lehmann is Director of Undergraduate Advising for the School of Math at Georgia Tech. In this position, he has a variety of duties, including running undergraduate math advising and coordinating multivariable calculus.

By the end of the semester, students should be able to meet the learning outcomes outlined below. I have categorized these learning outcomes by sections of our course content. Assessments throughout the semester may reference these learning outcomes.

Three-Dimensional Space, Graphs, and Vector Functions (G)

  • G1: Lines and Planes. I can describe lines using the vector equation of a line. I can describe planes using the general equation of a plane. I can find the equations of planes using a point and a normal vector. I can find the intersections of lines and planes. I can describe the relationships of lines and planes to each other. I can solve problems with lines and planes.

  • G2: Calculus of Curves. I can compute tangent vectors to parametric curves and their velocity, speed, and acceleration. I can find equations of tangent lines to parametric curves. I can solve initial value problems for motion on parametric curves.

  • G3: Geometry of Curves. I can compute the arc length of a curve in two or three dimensions and apply arc length to solve problems. I can compute normal vectors and curvature for curves in two and three dimensions. I can interpret these objects geometrically and in applications.

  • G4: Surfaces. I can describe, find, and graph traces and level curves of surfaces. I can identify standard quadric surfaces including: spheres, ellipsoids, elliptic paraboloids, hyperboloids, cones, and hyperbolic paraboloids. I can match graphs of functions of two variables to their equations and contour plots and determine their domains and ranges.

  • G5: Parameterization. I can find parametric equations for common curves, such as line segments, graphs of functions of one variable, circles, and ellipses. I can match given parametric equations to Cartesian equations and graphs. I can parameterize common surfaces, such as planes, quadric surfaces, and functions of two variables.

Differentiation in Higher Dimensions (D)

  • D1: Limits of Functions. I can calculate the limits of some functions of two variables or and apply the Two-Path Test to determine if they do not exist. I can state the definition of continuity for functions of multiple variables.

  • D2: Computing Derivatives. I can compute partial derivatives, total derivatives, directional derivatives, and gradients. I can use the Chain Rule for multivariable functions to compute derivatives of composite functions.

  • D3: Tangent Planes and Linear Approximations. I can find equations for tangent planes to surfaces and linear approximations of functions at a given point and apply these to solve problems.

  • D4:Optimization. I can locate and classify critical points of functions of two variables. I can find absolute maxima and minima on closed bounded sets. I can use the method of Lagrange multipliers to maximize and minimize functions of two or three variables subject to constraints. I can interpret the results of my calculations to solve problems.

Multiple Integration (I)

  • I1: Double & Triple Integrals. I can set up double and triple integrals as iterated integrals over any region. I can sketch regions based on a given iterated integral.

  • I2: Iterated Integrals. I can compute iterated integrals of two and three variable functions, including applying Fubini’s Theorem to change the order of integration of an iterated integral.

  • I3: Change of Variables. I can use polar, cylindrical, and spherical coordinates to transform double and triple integrals and can sketch regions based on given polar, cylindrical, and spherical iterated integrals. I can use general change of variables to transform double and triple integrals for easier calculation. I can choose the most appropriate coordinate system to evaluate a specific integral.

Vector Calculus (V)

  • V1: Line Integrals. I can set up and evaluate scalar and vector field line integrals in two and three dimensions.

  • V2: Conservative Vector Fields. I can test for conservative vector fields and find potential functions. I can state and apply the Fundamental Theorem of Line Integrals.

  • V3: Generalizations of the FTC. I can state and apply Green’s Theorem, Stokes’ Theorem and the Divergence Theorem to solve problems in two and three dimensions. I can choose which theorem is appropriate for different integrals.

  • V4: Surface Integrals. I can set up and compute surface integrals for scalar and vector valued functions.

Applications of Calculus (A)

  • A1: Interpreting Derivatives. I can interpret the meaning of a partial derivative, a gradient, or a directional derivative of a function at a given point in a specified direction, including in the context of a graph or a contour plot.

  • A2: Integral Applications. I can use multiple integrals to solve physical problems, such as finding area, average value, volume, or the mass or center of mass of a lamina or solid. I can interpret mass, center of mass, work, flow, circulation, flux, and surface area in terms of line and/or surface integrals, as appropriate.