I have grouped the standards into five larger categories: “geometry of vectors”, “functions, curves, and surfaces”, “differentiation”, “integration”, and “classical theorems”. In some SBG implementations, the competencies listed below might be more finely sorted out, but I have come to believe that giving college students a somewhat broader classification will encourage them to guide their own education and to think holistically about the material. I have tried to sort these according to three basic principles:

- they cover roughly the same amount of course material;
- they are roughly of the same importance towards mastering the content;
- they can be more-or-less independently measured (although there are indisputably dependences among them).

Despite these misgivings about the list itself, I feel it’s important to go ahead and publish it so that I and others can reflect on it. So here goes: (

*)*

**Update 1/18:**Following some feedback and discussion, I have revised this list from its original form. Please ignore the parts that have been struck out.__Geometry of vectors__

**Operations**– compute and interpret sum, scalar multiples, dot product, determinant, and cross product of vectors in the plane or in space**Objects**– describe points, lines, planes, spheres, and other surfaces using equations, vectors, set notation, or geometric objects of a different kind

__Functions, curves, and surfaces__

**Visualization**– sketch or predict appearance of the graph of a function or curve based on a formula or other description; sketch or describe level sets of a function; use computer software to examine shapes of graphs**Parametrization**(*added*) – find parametrizations of lines, circles and other curves, as well as planes, spheres, and other surfaces (e.g., tori and graphs of functions)**Analysis**– find domain and range of functions of 1, 2, and 3 variables, and describe these using set notation or geometric terminology; determine continuity

__Derivatives__

**Operators**– apply and interpret partial derivative, Jacobian, and gradient operators to functions; divergence and curl to vector fields**Operations on functions**(*added*) – compute and interpret partial derivatives and gradient of a function of 2 or 3 variables; apply and explain equality of mixed partial derivatives, including sufficient conditions for such equality to hold**Operations on vector fields**(*added*) – compute and interpret divergence and curl of a vector field**Linearization**– find tangent vectors, tangent lines, and tangent planes; use these to approximate curves and surfaces near a point**Higher derivatives**– apply and explain equality of mixed partial derivatives; use higher derivatives to collect data about shape of the graph of a function, including classifying critical points**Optimization**(*added*) – use higher derivatives to collect data about shape of the graph of a function; classify critical points; Lagrange multipliers**Differential equations**– interpret partial differential equations (such as the wave equation, the heat equation, and the Laplace equation) in terms of their solutions; verify that a given function is a solution to a PDE; use a computer to solve or approximately solve PDEs

__Integrals__

**Multiple integrals**– accurately describe regions over which double or triple integrals are computed; perform calculations of double or triple integrals; apply and justify change-of-coordinate formulas; use computers to find integrals**Line and surface integrals**–~~effectively parametrize curves and surfaces, and use these~~use parametrizations of curves and surfaces to compute length, area, work, and flux integrals**Applications**– use double and triple integrals to solve problems in geometry and probability; use line and surface integrals to investigate physical phenomena

__Classical theorems__

**Integrability conditions**– check conditions for a vector field to be irrotational or divergence-free (“closed” with respect to curl or divergence operators), and explain the meaning of these conditions; find potential functions**Generalizations of FTC**– explain the meaning and significance of Green’s Theorem, Divergence Theorem, and Stokes’ Theorem**Applications**– use Green’s Theorem, the Divergence Theorem, and Stokes’ Theorem to convert integrals between various forms**Generalizations of FTC**– explain the meaning and significance of Green’s Theorem, Divergence Theorem, and Stokes’ Theorem; use these to convert integrals between various forms

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