EM 127 - Mathematical Analysis II (Análise Matemática II) Introduction to surfaces in R3: quadric, cylindrical and revolution surfaces. General notions for scalar functions of n variables: domain and graph. Vector-valued functions of n variables; parametrical representation for curves in Rn and surfaces in R3. Introductory topological notions on Rn. Limits and continuity for scalar and vector-valued functions of n variables. Differentiation: partial and directional derivatives; gradient vector; partial derivatives of higher order; total derivative or Fréchet derivative and differentiability of a scalar function of n variables. Applications of the gradient: tangent plane and maximum of a directional derivative. Differentiability of vector-valued functions of n variables–Jacobian matrix. Properties of the derivative; different cases of the chain rule. Functions defined implicitly; implicit function theorem and implicit differentiation. Taylor’s formula for scalar functions of n variables. Extrema of scalar functions of n variables; constrained extrema and Lagrange multipliers. Integration: complements of Riemann integral for real-valued functions of one variable – improper integrals. Double integrals: over a rectangle and over more general regions in R2. Properties and geometric interpretation of double integrals. Fubini theorem – changing the order of integration. Applications of double integrals to areas and volumes, average values, center of mass and moment of inertia. Changing variables in double integrals; double integrals in polar coordinates. Triple integrals: over rectangular parallelepiped and more general regions in R3 . Properties and geometric interpretation of triple integrals. Fubini theorem – changing the order of integration for triple integrals. Applications to volumes, average values, center of mass and moment of inertia. Changing variables: triple integrals in cylindrical and spherical coordinates.. <<<<<<<<<<  BACK  <<<<<<<<<<