If you successfully complete this course, you will be able to:  

In Vectors and the Geometry of Space

• • Identify and apply the parts of the three-dimensional coordinate system, distance formula and the equation of the sphere;
  • Compute the magnitude, scalar multiple of a vector, and find a unit vector in the direction of a given vector;
  • Calculate the sum, difference, and linear combination of vectors;
  • Calculate the dot product and cross product of vectors, use the products to calculate the angle between two vectors, and to determine whether vectors are perpendicular or parallel;
  • Determine the scalar and vector projections;
  • Write the equations of lines and planes in space;
  • Draw various quadric surfaces and cylinders using the concepts of trace and cross-section.

In Vectors Functions

• • Sketch vector-valued functions;
  • Determine the relationship between these functions and the parametric representations of space curves;
  • Compute the limit, derivative, and integral of a vector-valued function;
  • Calculate the arc length of a curve and its curvature; identify the unit tangent, unit normal and binormal vectors;
  • Calculate the tangential and normal components of a vector;
  • Describe motion in space.

In Partial Derivatives

• • Define functions of several variables and know the concepts of dependent variable, independent variables, domain and range;
  • Calculate limits of functions in two variables or prove that a limit does not exist;
  • Test the continuity of functions of several variables;
  • Calculate partial derivatives and interpret them geometrically, calculate higher partial derivatives;
  • Determine the equation of a tangent plane to a surface;
  • Calculate the change in a function by linearization and by differentials;
  • Determine total and partial derivatives using chain rules;
  • Calculate directional derivatives and interpret the results o Identify the gradient, interpret the gradient, and use it to find directional derivative;
  • Apply intuitive knowledge of concepts of extrema for functions of several variables, and apply them to mathematical and applied problems.
  • Understand method of Lagrange multipliers.

In Multiple Integrals

• • Define double integral, evaluate a double integral by the definition and the midpoint rule and describe the simplest properties of them;
  • Calculate iterated integrals by Fubini's Theorem;
  • Calculate double integrals over general regions and use the geometric interpretation of double integral as a volume to calculate such volumes. Some applications of double integrals may include computing mass, electric charge, the center of mass and moment of inertia;
  • Evaluate double integrals in polar coordinates to calculate polar areas, evaluate Cartesian double integrals of a particular form by transforming to polar double integrals;
  • Define triple integrals, evaluate triple integrals, and know the simplest properties of them. Calculate volumes by triple integrals;
  • Transform between Cartesian, cylindrical, and spherical coordinate systems; evaluate triple integrals in all three coordinate systems; make a change of variables using the Jacobian.

In Vector Calculus

• • Describe vector fields in two and three dimensions graphically; determine if vector fields are conservative, directly and using theorems;
  • Identify the meaning and set-up of line integrals and evaluate line integrals;
  • Apply the connection between the concepts of conservative force field, independence of path, the existence of potentials, and the fundamental theorem for line integrals;
  • Calculate the work done by a force as a line integral;
  • Apply Green's theorem to evaluate line integrals as double integrals and conversely;
  • Calculate and interpret the curl, gradient, and the divergence of a vector field;
  • Evaluate a surface integral;
  • Understand the concept of the flux of a vector field;
  • State and use Stokes Theorem o State and use the Divergence Theorem.