| Competences |
| Type A | Code | Competences Specific |
| FB1 | Have the ability to solve mathematical problems that may arise in engineering. Have the ability to apply knowledge on: linear algebra; geometry, differential geometry, differential and integrated calculation, differential equations and partial derivatives, numerical methods, numerical algorithmics, statistics and optimisation. | |
| Type B | Code | Competences Transversal |
| B2 | Have knowledge in basic and technological subjects, which gives them the ability to learn new methods and theories, and the versatility to adapt to new situations. | |
| Type C | Code | Competences Nuclear |
| Learning outcomes |
| Type A | Code | Learning outcomes |
| FB1 |
Understand the genesis and fundamentals of ordinary differential equations. Solve differential equations of the first order. Know the notion of equation characteristic of a linear differential equation with constant coefficients. Solve linear differential equations of the second order with constant coefficients. Know methods for mathematically modelling physics and technology problems. Understand the notions of limit and continuity of a real function of diverse variables. Know the concept of contour lines and surfaces. Understand the concept of directional derivative of a real function of different variables. Understand the concept of the Jacobian matrix. Understand the concept of gradient of a real function of different variables. Solve problems of limits, continuity and derivability of a real function of diverse variables. Understand the concept of differential of a real function of different variables. Understand the concept of tangent space and normal straight line to a surface at a point. Analyse whether a function can be differentiated. Solve problems of optimisation related to functions of diverse variables. Understand the concepts of double and triple integral in terms of geometry and shape. Understand the fundamentals of PDEs. | |
| Type B | Code | Learning outcomes |
| B2 |
Know methods for mathematically modelling physics and technology problems. Solve problems of optimisation related to functions of diverse variables. Understand the concepts of double and triple integral in terms of geometry and shape. | |
| Type C | Code | Learning outcomes |
| Contents |
| Topic | Sub-topic |
| Multivariable Calculus | - 1. Function domain and range. Function operations. Composition of functions. Level curves. Conics. - 2. Limits and continuity. Directional derivative and partial derivative - 3. Differential and Gradient. Tangent plane and normal line. - 4. Partial Derivatives of Higher order. Taylor formula. Implicit functions. - 5. Local minimum and maximum. Relative extremes. Hessian. Absolute extremes in a compact domain. - 6. Conditional extremes. Lagrange multipliers. |
| Ordinary Differential Equations | - 11. Ordinary Differential Equations (ODEs). Order. Solution of a ODE. Existence and unicity of the solution. - 12. ODEs of First Order. Separation of Variables. Exact Differential. Integrating Factor. Linear Equations. - 13. Second Order ODEs. Particular Solution. Reduction of Order. Lineal ODEs with constant coefficients. Homogeneous and no homogeneous equations. Method of the indeterminate constants. - 14. Models. Population. Disintegration. Heat transfer. Oscillations. |
| Multiple Integration |
- 7. Double Integral. Properties. Geometric Interpretation. Fubini's Theorem. - 8. Change of variables. Jacobian of the transformation. Polar coordinates. Volume computation. - 9. Triple Integral. Properties. Cylindrical and Spherical coordinates. - 10. Center of Gravity. Inertia moments. |
| Planning |
| Methodologies :: Tests | ||||||||||
| Competences | (*) Class hours |
Hours outside the classroom |
(**) Total hours | |||||||
| Introductory activities |
|
1 | 1 | 2 | ||||||
| Lecture |
|
28 | 56 | 84 | ||||||
| Practical cases/ case studies |
|
14 | 14 | 28 | ||||||
| Problem solving, exercises in the classroom |
|
26 | 0 | 26 | ||||||
| Personal attention |
|
1 | 0 | 1 | ||||||
| Extended-answer tests |
|
3 | 0 | 3 | ||||||
| Practical tests |
|
2 | 4 | 6 | ||||||
| (*) On e-learning, hours of virtual attendance of the teacher. (**) The information in the planning table is for guidance only and does not take into account the heterogeneity of the students. | ||||||||||
| Methodologies |
| Description | |
| Introductory activities | Subject presentation |
| Lecture | Content development |
| Practical cases/ case studies | Problem statement |
| Problem solving, exercises in the classroom | Resolution of exercises that have been raised previously |
| Personal attention |
| Personalized attention |
| Description |
|
Personal attention in the classroom, helping to solve the exercises. Personal attention in the office to resolve doubts. |
| Assessment |
| Methodologies | Competences | Description | Weight | ||||
| Practical tests |
|
A global test on resolution of problems on the topics of the course. It is imperative to obtain a qualification of at least 3/10 points in order to be able to obtain the grade of this course. |
40% | ||||
| Extended-answer tests |
|
Two additional tests practically oriented where the students will need to solve problems and exercises on the topics of the course. Each of these tests will have a 30% of the final qualification These additional tests do not need to be necessarily standard exams. The professors my decide to use any other tool, web based or not, that they consider appropriate to obtain an accurate and personal qualification of the students. |
60% | ||||
| Others |
| Other comments and second exam session | |||
|
The Second evaluation will consist in a single practical exam with problems based on the different topics of the course. |
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| Sources of information |
| Basic |
Piskunov, Calculo diferencial y integral, Ed Paraninfo,
Jon Rogawski, Cálculo (Varias variables), Ed. Reverté, 2012
Dennis G.Zill et altres, Cálculo de Varias variables, McGraw-Hill, 2016 |
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| Complementary | |||||||||
| Recommendations |
| Subjects that it is recommended to have taken before | |||
|
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(*)The teaching guide is the document in which the URV publishes the information about all its courses. It is a public document and cannot be modified. Only in exceptional cases can it be revised by the competent agent or duly revised so that it is in line with current legislation.