- Describe vectors in the plane, and write them in component form and in terms of basis vectors.
- Compute the magnitude of a vector in 2-space.
- Describe and perform basic operations on vectors in 2-space.

- Distinguish 3-dimensional space from 2-dimensional space.
- Work with the rectangular coordinate system in 3-space.
- Perform and describe basic operations on vectors in 3-space.
- Compute the magnitude of a vector in 3-space.

- Compute the dot product of two vectors.
- Find and use the projection of one vector onto another.
- Find the direction cosines of a vector.
- Use the dot product in applications involving orthogonality, work, angles between vectors, projections, etc.

- Compute the cross product of two vectors.
- Find a vector orthogonal to two given vectors.
- Use the cross product in applications of area, volume, and torque.

- Find parametric or symmetric equations for a line in space.
- Find the equation of a plane in space.
- Find the distance from a point to a line.
- Find the distance from a point to a plane.
- Find the angle between two planes.

- Identify cylinders as 3-dimensional surfaces.
- Recognize quadric surfaces from their graphs or equations.
- Roughly sketch the graph of a quadric surface.
- Sketch the traces of a quadric surface.

- Convert among rectangular, cylindrical, and spherical coordinates.

- Recognize and use equations for vector-valued functions.
- Sketch and/or describe the graph of a vector-valued function.
- Find limits of vector-valued functions.
- Determine the points of continuity of a vector-valued function.

- Evaluate and use the derivative of a vector-valued function.
- Compute the unit tangent vector for a vector-valued function.
- Integrate vector-valued functions.

- Determine the length of a plane or space curve defined by a vector-valued function.
- Find the arc-length parameterization for a smooth curve.
- Find the curvature of a smooth curve at a point.
- Compute the principal unit normal vector for a smooth curve.

- Find the tangential and normal components of acceleration.
- Solve a projectile motion problem in space.

- Recognize a function of two variables and identify its domain and range.
- Sketch or describe the graph of a function of two variables.
- Recognize a function of three or more variables and identify its domain and range.
- Identify and describe the level curves or surfaces of a function of several variables.

- Compute the limit of a multi-variable function.
- Use the two-path test to show that a limit does not exist.
- Determine the points of discontinuity of a function of several variables.

- Compute the partial derivatives of a function of several variables.
- Compute higher-order partial derivatives.
- Describe the conditions for equality of higher-order mixed partial derivatives.
- Interpret partial derivatives as slopes.

- Determine if a function of two variables is differentiable.
- Compute the total differential of a function and use it to approximate change.
- Find an equation of the plane tangent to a given surface at a point.
- Find parametric equations for the line normal to a given surface at a point.
- Use tangent planes (i.e., linearizations) to approximate function values.

- State and use the chain rules for any number of independent and intermediate variables.
- Use implicit differentiation.

- Compute directional derivatives and interpret them as slopes.
- Compute gradient vectors.
- Use gradient vectors as normal vectors.
- Use gradient vectors to determine the direction of maximum increase/decrease.

- Find the critical points of a function of two variables.
- Use the second partials test to classify critical points.
- Use critical points and boundary points to find the absolute extrema for functions of two variables.

- Use Lagrange multipliers to solve a constrained optimization problem.

- Write a double integral as an iterated integral and evaluate.
- Use a double integral to compute the area of a region, the volume under a surface, or the volume between two surfaces.
- Change the order of integration.

- Evaluate a double integral in polar coordinates.
- In applications of double integrals, convert from rectangular to polar coordinates or vice versa.
- Use double integrals in polar coordinates to compute areas and volumes.

- Write a triple integral as an iterated integral and evaluate.
- Use a triple integral to compute the volume of a space region.
- Use a triple integral to compute the average value of a function over a space region.
- Change the order of integration.

- Evaluate a triple integral by converting to cylindrical coordinates.
- Evaluate a triple integral by converting to spherical coordinates.

- Use a triple integral to find the mass of a solid in space.

- Recognize a vector field in the plane or in space.
- Identify a conservative vector field and find an associated potential function.

- Evaluate line integrals.
- Use line integrals in applications.

- Use the Fundamental Theorem of Line Integrals to compute the line integral of a conservative vector field.

- Apply Green's theorem.

*Last updated December 7, 2022*