Frequency: Fall/Spring Terms 

Credit: (4-2)5 

Content: Complex numbers. Vectors, lines and planes in space, scalar and vector products. Vector valued functions. Space curves. Functions of several
   variables: Limit, continuity, partial derivative, directional derivative. Tangent plane. Extreme values. Method of Lagrange multipliers. Multiple integrals.
   Cylindrical and spherical coordinates. Line, surface integrals. Green's Theorem. Gauss' and Stokes' Theorems.  

Goals: Calculus was first discovered to meet the needs of the scientists of the sixteenth and seventeenth centuries. Diferential calculus deals with the problem of calculating rates of change. It enables us to define slope of curves, to calculate velocities, accelerations of moving bodies and to predict the times when planets would be closest together or farthest apart. Integral calculus deals with the problem of determining a function from information about its rate of change. It enables us to calculate the future location of a body from its present position, to find the areas of irregular regions in the plane, to measure the lengths of curves, and to find the volumes and masses of arbitrary solids. The goal of this course is to present a modern view of calculus enhanced by the use of technology.  

Course Outline: 

•    (1 Week) Complex Numbers  
•    Conic (only Classifying General Conics)  
•    Analytic Geometry in Three and More Dimensions  
•    Plane Vectors (omit Hanging Cables and Weights) 
•    (1 Week) Vectors in 3-Space  
•    The Cross Product in 3-Space (omit Determinants) 
•    Planes and Lines  
•    Quadric Surfaces  
•    (1 Week) Functions of Several Variables  
•    Limits and Continuity  
•    Partial Derivatives(omit Distance from a Point  to a Surface: a Geometric Example) 
•    Higler-Order Derivatives  
•    (1 Week) The Chain Rule (omit Homogeneous Functions) 
•    Linear Approximations, differentiability, and differentials (omit Proof of the Chain Rule)  
•    Review 
•    (1 Week) Gradients and Directional Derivatives  
•    Implicit Functions  
•    Taylor Series and Approximations  (omit Approximating Implicit Functions) 
•    (1 Week) Extreme Values  
•    Extreme Values of Functions Defined on Restricted Domains  (omit Linear Programming) 
•    Lagrange Multipliers  
•    Parametric Problems  (only Differentiating Integrals with Parameters) 
•    (1 Week) Double Integrals  
•    Iteration of Double Integrals in Cartesian Coordinates  
•    Improper Integrals and a Mean Value Theorem  
•    Double Integrals in Polar Coordinates  
•    (1 Week) Review 
•    (1 Week) Review 
•    (1 Week) Triple Integrals  
•    Change of Variables in Triple Integrals  
•    Vector Functions of One Variable  
•    Curves and Parametrizations  
•    (1 Week) Curvature, Torsion, and the Frenet Frame   (up to Torsion and Binomial, the Frenet-Serret Formulas) 
•    Curvature and Torsion for General Parametrizations  (up to Evaluates) 
•    Vector and Scalar Fields  (up to Field Lines (Integral Curves) 
•    Conservative Fields (up to Sources, Sinks, and Dipoles) 
•    (1 Week) Line Integrals  
•    Line Integrals of Vector Fields  
•    Surfaces and Surface Integrals  (omit the Attraction of a Spherical Shell) 
•    Surface Integrals of Vector Fields  
•    Gradient, Divergence, and Curl (up to Interpretation of the Divergence) 
•    (1 Week) Some Identities Involving Grad, Div., and Curl  
•    Review 
•    (1 Week) Green’s Theorem and Stokes’s Theorem  
•    Review 

 

Reference Books: 

• G.B. Thomas and R.Finney; Calculus and Analytic Geometry. Addison-Wesley Publishing Com. 1996.  
• R.Ellis and D. Gulick; Calculus with Analytic Geometry, 5 th Edition, SCP, 1994  
• M.Spivak; Calculus 3 rd Edition, Publish of Perish, Inc.