Multi-Variable Calculus Course Description

Multi-Variable Calculus course covers topics found in a typical semester-long multivariable calculus course at universities.
Advanced Topic Multivariable Calculus will include visualizing and working with functions of several variables, vectors and vector-valued functions, differentiating functions of several variables, gradients, partial derivatives, and multiple integration of several variables. The Advanced Topic (AT) designation indicates a course is at university level, putting it at or above the level of a traditional Advanced Placement(AP) course. The course requires rigorous study and emphasizes in-depth research.

Syllabus

1. Vectors, the Geometry of Space, Vector Functions
   -Use vector operations, including dot and cross products, to solve geometric problems in three-dimensionalspace
   -Derive and apply equations for planes and quadric surfaces, including tangent and normal planes, and identify surfaces from their symbolic representations.
   -Analyze the properties of space curves using vector-valued functions, including calculating unit tangent and normal vectors, curvature, and the equations of osculating planes, or applying these to model projectile motion in a gravitational field.
   
2. Partial Derivatives
   -Calculus of Multivariable Functions: Evaluate the existence of limits for functions of several variables by testing different approach paths; calculate and interpret partial derivatives, including higher-order derivatives and the Chain Rule, to find rates of change and approximate function changes.
   -Gradients and Directional Analysis: Utilize the gradient vector to calculate directional derivatives, identify the direction of maximum increase for a potential function, and analyze contour maps to determine gradient properties.
   -Optimization and Extrema: Identify and classify local extrema and saddle points using the Second Derivatives Test and solve constrained optimization problems using Lagrange multipliers to find absolute maximum and minimum values.
   
3. Multiple Integrals
   -Evaluate double and triple integrals in rectangular, polar, cylindrical, and spherical coordinate systems to calculate area, volume, surface area, or mass.
   -Apply the Jacobian of a transformation to change variables in multiple integrals to simplify the evaluation of complex regions.
   
4. Vector Calculus
   -Compute the curl and divergence of vector fields and determine if a vector field is conservative by finding its potential function.
   -Apply and verify the major theorems of vector calculus—including the Fundamental Theorem for Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem—to evaluate line and surface integrals.