## Mathematics II

To solve complicated problems in Mechanical Engineering, such as the ones that are expected to influence technological evolution in the near future, is a process that demands careful planning and logical precision, in other words, scientific treatment, and, therefore, it cannot be achieved only by means of vocabular reasoning. Mathematics is the mean for encoding the various problems involved in engineering processes, thus resulting in a most valuable tool in the hands of Mechanical Engineers. A first encounter of our students with the realm of logic and numbers has already taken place in the context of the course Mathematics I. In the second semester, the students of our Department are going to deal with more sophisticated mathematical concepts, such as calculus of functions of multiple variables and differential equations (Mathematics II).

### Objectives

After successfully completing the course, students of the Department of Mechanical Engineering TE it must: The Calculus of Several Variables: - To know the basic concepts related to functions of two or / and more variables (place definition, geometric interpretation, etc.). - To know in depth the concept of partial derivative first and second class as well as the corresponding "mixed" derivative. - To solve problems of partial derivation of composite and implicit functions, as well as difficulties in determining the total differential. - To extremities face value problems of functions of several variables by means of partial derivatives (minimum and maximum "saddle" spots). - To know the fundamentals of vector analysis and concepts of gradient, divergence and rotation of vector fields, with particular emphasis on the quantitative development of these aggregates. - Be able to calculate the price of double integrals in Cartesian and / or polar coordinates, and the quantities involved in their applications (rigid body mass, moments of inertia, etc.). Differential Equations: - To know the basic concepts of first order differential equations (general and partial solution, initial conditions). - To solve different types of differential first-order equations - differential equations with separated variables and stemming them, homogeneous differential equations and stemming them, linear differential equations of first order, complete differential equations using (or not) integral factor . - Be able to deal with problems of Physics and Applied Technology, the composition and resolution of the first order differential equation that describes them. - To adequately manage linear differential second order equations with constant coefficients and non-zero second member, with emphasis on the basic characteristics (homogeneous and complete differential equation groups of homogeneous solutions, selecting the partial termination of full depending on the functional expression of the second State initial and boundary conditions). - Be able to deal with problems of Physics and Applied Technology, the composition and resolution of the second-order differential equation that describes them.

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### Syllabus

Multivariable Calculus: Functions of two variables: Place definition and geometric interpretation. Differential equations: First and second-order, mixed derivative. Partial derivatives of composite and implicit functions. Total differential function of two variables. Extreme values ​​of functions of two variables - maxima, minima and "saddle" points. Vector Analysis: Vector fields - gradient, divergence and rotation. Double integrals: Place integration, geometric interpretation. Resolving double integrals in Cartesian and polar coordinates. Applications of double integrals - rigid body mass, moments of inertia. Differential Equations: First order differential equations: General and partial solution, initial conditions. Species first order differential equations - Differential equations with separated variables and stemming them, homogeneous differential equations and stemming them, linear differential equations of first order, complete differential equations, integral factors. Physical and Technological Applications of first order differential equations. Second order linear differential equations with constant coefficients and non-zero second member: Main characteristics - homogeneous and complete differential equation, solutions of homogeneous categories, selecting the partial solution of the full, initial and boundary conditions.

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