Class Mathematics II

Analytic geometry and differential and integral calculus for real functions of several real variables.

1. Vectors and space geometry: 1.1 Three-dimensional Cartesian coordinates. 1.2 Vectors, dot product. 1.3 Lines and planes. 1.4 Cylindrical and quadric surfaces. 1.5 Topological notions

2. Real variable vector functions: 2.1 Vector functions and curves in space. 2.2 Limits and continuity. Differentiability, vector tangent to a curve. 2.3 Integration, length of a curve.

3. Functions of several variables: 3.1 Domains and contour lines. 3.2 Limits and continuity. 3.3 Partial derivatives, higher order derivatives. 3.4 Differentiability, tangent planes and polynomial approximations. 3.5 Directional derivative, gradient vector and its geometric interpretation. 3.6 Hessian matrix, local extremes and saddle points.

4. Multiple integrals: 4.1 Double integrals: 4.2 Polar coordinates and change of variables in double integrals. 4.3 Triple integrals. 4.4 Cylindrical and spherical coordinates. Changing variables in triple integrals.

• In class, the ideas underlying the curriculum of this course are discussed, and multiple examples and application exercises are analyzed.
• For each topic in this course, a set of application exercises is presented. Students are encouraged to solve these exercises and to raise any doubts they may have.
• All supporting materials and relevant information will be shared with students through Moodle.
• The assessment includes a continuous component, which involves completing three 40-minute tests and a final exam (Final Exam or Make-up).
  • The average of the three tests is denoted as A, and the Exam Grade is denoted as B.
  • If A > 9.5, the student is approved for the course and can take the exam to improve the grade. In this case, the Final Grade = max(A, B).
  • If A < 9.5, the student is not approved for the course and must take the exam to obtain approval.
• Students who achieve a final grade of at least 10 points are considered approved.

• Stewart, J.; Cálculo, vol. 2, 5ª ed.; Thomson Learning; 2007.

• Simmons, G.F.; Cálculo com Geometria Analítica, vol. 2; Makron Books; 1987.