Description

Calculus III is the third and the final part of our standard three-semester calculus sequence. The distinct feature of this part of the course is its focus on the multi-dimensional analysis, as opposed to one-dimensional analysis that students learned in Calculus I and Calculus II.

Calculus III focuses on important concepts such as that of a vector, a vector field, geometry of space,a function of several variables, partial derivative, a line-integral and application of the line integral, including Green’s Theorem, the Divergence Theorem, and the Stokes’ Theorem.. The ideas of the vector calculus apply to numerous areas of human knowledge such as engineering, physics, chemistry, pure mathematics, biology, and many others.

Prerequistie: Calculus I

Intended Learning Outcomes

After completing this course, student should be able to:

• extend single variable calculus concepts to higher dimensions (e.g. partial derivatives, gradients, integrals, etc.).
• introduce vector notation and algebra.
• compute dot product, cross product, length of vectors.
• explore visualizations in space.
• use Calculus to compute quantities from physics such as: motion of a particle (velocity, acceleration, distance travelled); mass, center of mass; work.
• introduce parameterizations of curves and surfaces.
• introduce particle motion concepts (e.g. velocity and acceleration)
• introduce arc length and curvature.
• introduce multivariate optimization.
• introduce non-rectangular coordinate systems.
• evaluate integrals of functions or vector-related quantities over curves, surfaces, and domains in two- and three-dimensional space
• introduce generalizations of the fundamental theorem of calculus (e.g. Green’s theorem).

Course Contents

Ch. 12.  Vectors and the Geometry of Space.

1. Three-Dimensional Coordinate Systems.
2. Vectors.
3. The Dot Product.
4. The Cross Product.
5. Lines and Planes in Space.
6. Cylinders and Quadric Surfaces .

Ch. 13.  Vector-Valued Functions and Motion in Space. 

1. Vector Functions.
2. Modeling Projectile Motion.
3. Arc Length and the Unit Tangent Vector T.
4. Curvature and the Unit Normal Vector N.
5. Torsion and the Unit Binormal Vector B.
6. Planetary Motion and Satellites.

Ch. 14.  Partial Derivatives. 

1. Functions of Several Variables.
2. Limits and Continuity in Higher Dimensions.
3. Partial Derivatives.
4. The Chain Rule.
5. Directional Derivatives and Gradient Vectors.
6. Tangent Planes and Differentials.
7. Extreme Values and Saddle Points.
8. Lagrange Multipliers.
9. *Partial Derivatives with Constrained Variables.
10. Taylor's Formula for Two Variables.

Ch. 15.  Multiple Integrals. 

1. Double Integrals.
2. Areas, Moments and Centers of Mass*.
3. Double Integrals in Polar Form.
4. Triple Integrals in Rectangular Coordinates.
5. Masses and Moments in Three Dimensions.
6. Triple Integrals in Cylindrical and Spherical Coordinates.
7. Substitutions in Multiple Integrals.

 Ch. 16. Integration in Vector Fields. 

1. Line Integrals.
2. Vector Fields, Work, Circulation, and Flux.
3. Path Independence, Potential Functions, and Conservative Fields.
4. Green's Theorem in the Plane.
5. Surface Area and Surface Integrals.
6. Parametrized Surfaces.
7. Stokes' Theorem.
8. The Divergence Theorem and a Unified Theory.

Recommended Books

• Thomas, Calculus, 11th Edition. Pearson. (Ch 12 - 16)
•  Anton H., Bevens I., Davis H., Calculus, latest Edition, John Wiley & Sons, Inc.
• Larson E. Calculus, latest edition,  Brooks/Cole Cengage Learning

System of Evaluation

Sessional: 20 (Presentation  15, Attendance 05, )

Mid-Term Exam:  30 (Multiple choice questions)

Final-Term Exam: 50 (Multiple choice questions)

Key Dates and Time of Class Meeting

 Class                                       (BS- III (R+SS))

Tuesday                                     03:00 PM- 05:00 PM 

Wednesday                               10:00 AM - 11:00 AM 

Thursday                                   10:00 AM - 11:00 AM 

Commencement of Classes      October 12, 2020

Mid Term Examination               December14 - 18, 2020

Final Term Examination             Febraury 08 - 12, 2021

Declaration of Result                 Febraury 19, 2021