Upon successful completion of the course the student will be able to:
1. Extend and apply algebraic and geometric concepts of two dimensional vectors in the Cartesian plane
to 3-dimensions, including the distance between vectors, vector algebra, and the Euclidean norm of a
vector.
2. Apply operations involving the inner product, the cross product, and triple scalar product of
3-dimensional vectors and use these operations in geometric and physical applications.
3. Calculate the angle between vectors, and determine if two vectors are orthogonal.
4. Set up the equation of the line in both vector and parametric form, and the equation of a plane in
3-space, and calculate the distances between points, planes and lines.
5. Recognize, compare, contrast, and sketch the different quadric surfaces.
6. Implement changes of variables between rectangular, cylindrical, and spherical coordinates.
7. Sketch simple single variable vector-valued functions in R^2 and R^3.
8. Compute the limit, derivative, and integrals of vector-valued functions of one variable.
9. Determine and identify the continuity of a single variable vector-valued function at a single point and
throughout a set.
10. Compute the unit tangent vector, principal unit normal vector, the arc length and the curvature of a
vector-valued function.
11. Design and apply some elementary concepts in point set topology as they relate to sets in
multi-dimensions.
12. Describe and apply the formal definitions of limits, and continuity from single variable calculus to
functions of 2, 3 and n-variables.
13. Calculate first as well as higher order partial derivatives of multivariable functions.
14. Define the derivative and the concept of the differentials of multivariable functions, and calculate
linear and quadratic approximations to multivariable scalar functions.
15. Apply the Chain Rule to a composition of multivariable functions.
16. Calculate the directional derivative of a multivariable function at a point in a given direction; and
the gradient of such a function, applying the properties of the gradient to describe the behavior of the
function.
17. Calculate the critical points of a differentiable multivariable function in an open ball, and applying
the second derivative test, determine if these points are relative maxima, relative minima or saddle
points.
18. Calculate the derivative of multivariable functions expressed implicitly by an equation, as well as
the derivative of inverse functions.
19. Demonstrate use of Lagrange's Theorem to compute the extrema of a multivariable function subject
to given constraints.
20. Calculate double and triple integrals over rectangular and non-rectangular regions, by iterating, by
changing the order of integration, or by changing variables.
21. Determine areas, volumes, surface area, mass, centers of mass, and moments of inertia.
22. Sketch a vector field and compute its curl and divergence.
23. Compute the line integral of a vector-valued function over a piecewise smooth contour.
24. Calculate the work done by a vector-valued multivariable function over a piecewise smooth contour.
25. Apply the concept of path independence and determine if a vector field is conservative, and if so,
calculate its potential energy function.
26. Apply Green's, Stokes' and the Divergence Theorems, and calculate surface integrals over
parametrized piecewise smooth surfaces to compute flux of a vector field.