Calculus 2

This course is the third part in the sequence the calculus courses. The course covers infinite series, vectors and multiple variable functions. Multivariable Calculus is the extension of Calculus to more than one variable. That is, in single variable calculus we study functions of a single independent variable, and in multivariable calculus we study functions of two or more independent variables. These functions are interesting in their own right, but they at the same time they have a lot for applications in the real world problems.

Course structure: The content of the course is organized into 5 (five) major units. The Midterm Exam covers the topics of the first two units. The Final Exam is intended to check the knowledge and ability of students on the last three units:

1. Infinite Sequences & Series. Taylor Series.

2. Vectors and Vector Calculus. Analytical Geometry.

3. Partial Derivatives

4. Double Integrals and Line Integrals in the Plane

5. Triple Integrals and Surface Integrals in 3-Space

• 3.1 | Alternating Series. Absolute convergence

• 3.2 | Comparison & Limit Comparison Tests

• 3.3 | Ratio Test of Convergence

• 3.4 | Root Test of Convergence

• 4.1 | Power Series: Interval & Radius of Convergence

• 4.2 | Taylor Series. Maclaurin Series

• 4.3 | Applications of Taylor Series

• 5.1 | Points in 3D

• 5.2 | Add & subtract vectors geometrically

• 5.3 | Create Position Vectors

• 5.4 | 4 Operations with Vectors

• 5.5 | Make Unit Vectors. Standard Unit Vectors.

• 6.1 | Dot product

• 6.2 | Angles between two vectors

• 6.3 | Projection vectors

• 6.4 | Cross product. Its geometric meaning.

• 6.5 | Equations on lines

• 7.1 | Dot product vs. Cross product

• 7.2 | Vector equation of Planes

• 7.3 | Scalar vs. Vector equation of planes

• 7.4 | Planes through 3 points

• 7.5 | Distance between a point and a plane

• 7.6 | Intro to Vector Functions

• 7.7 | Derivatives of Vector Functions

• 7.8 | Tangent Vector and Tangent Line

• 8.1 | Arc Length of a Vector Function

• 8.2 | Tangent, Normal & Binormal Vectors

• 8.3 | Curvature

• 8.4 | Calculating the Curvature

• 9.1 | Intro to Partial Derivatives

• 9.2 | Tangent Planes

• 9.3 | Multivariable Chain Rule

• 9.4 | Directional Derivatives

• 10.1| Gradient Vector shows the Steepest Descent

• 10.2| Max/Min values. Find the Saddle points.

• 11.1 | Double Integral over Rectangular Areas

• 11.2 | Double integral over general areas

• 11.3 | Double integrals in polar coordinates

• 12.1 | Surface Area using Double Integrals

• 12.2 | Jacobian 2D: change of variables

• 12.3 | Triple integration

• 12.4 | Jacobian 3D: change of variables

• 13.1 | Line Integrals of Scalar Functions

• 13.2 | intro to Vector Fields

• 13.3 | Line Integrals in Vector Fields

• 13.4 | Conservative Vector Fields