Acquire the fundamental concepts of differentiation and integration for multi-variable functions
teacher profile teaching materials
1. Functions of n real variables
Vector spaces. Scalar product (Cauchy-Schwarz inequality), norm, distance,
standard topology, compactness in Rn.
Continuous functions from Rn to Rm. Continuity and uniform continuity. Weierstrass theorem.
Definitions of partial and directional derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.
2. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces Teo. 6:10
3. Implicit functions
The theorem of implicit and Inverse functions.
Constrained maxima and minima, Lagrange multipliers.
4. Ordinary differential equations
Examples: equations with separable variables, linear systems with constant coefficients
(solution with matrix exponential).
Existence and uniqueness theorem.
Linear systems, structure of solutions, wronskian, variation of constants.
Fruizione: 20402076 AM210 - ANALISI MATEMATICA 3 in Matematica L-35 N0 PROCESI MICHELA, FELICI FABIO
Programme
1st semester:1. Functions of n real variables
Vector spaces. Scalar product (Cauchy-Schwarz inequality), norm, distance,
standard topology, compactness in Rn.
Continuous functions from Rn to Rm. Continuity and uniform continuity. Weierstrass theorem.
Definitions of partial and directional derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.
2. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces Teo. 6:10
3. Implicit functions
The theorem of implicit and Inverse functions.
Constrained maxima and minima, Lagrange multipliers.
4. Ordinary differential equations
Examples: equations with separable variables, linear systems with constant coefficients
(solution with matrix exponential).
Existence and uniqueness theorem.
Linear systems, structure of solutions, wronskian, variation of constants.
Core Documentation
Analisi Matematica II, Giusti - Analisi Matematica II, ChierchiaType of delivery of the course
4 hours of frontal teaching 2 of exercises and 2 of tutoring a week In the event of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities will be implemented. In particular, the following methods will apply: live remote lesson and recording of the lesson itself.Type of evaluation
The written exam focuses on the topics covered in class and tends to verify the ability to solve exercises. It consists of 3-4 exercises on the topics discussed during the class. The oral exam verifies the ability to present and prove the theorems studied in class and apply them in specific cases. In the event of an extension of the health emergency from COVID-19, all the provisions governing the methods of carrying out the assessment activities will be implemented. In particular, the following methods will apply: Assignment of exercises during the course that the student will have to present to access the oral exam, which will be held in remote mode. teacher profile teaching materials
Vector spaces. Scalar product (Cauchy-Schwarz inequality), norm, distance,
standard topology, compactness in Rn.
Continuous functions from Rn to Rm. Continuity and uniform continuity. Weierstrass theorem.
Definitions of partial and directional derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.
2. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces Teo. 6:10
3. Implicit functions
The theorem of implicit and Inverse functions.
Constrained maxima and minima, Lagrange multipliers.
4. Ordinary differential equations
Examples: equations with separable variables, linear systems with constant coefficients
(solution with matrix exponential).
Existence and uniqueness theorem.
Linear systems, structure of solutions, wronskian, variation of constants.
Fruizione: 20402076 AM210 - ANALISI MATEMATICA 3 in Matematica L-35 N0 PROCESI MICHELA, FELICI FABIO
Programme
1. Functions of n real variablesVector spaces. Scalar product (Cauchy-Schwarz inequality), norm, distance,
standard topology, compactness in Rn.
Continuous functions from Rn to Rm. Continuity and uniform continuity. Weierstrass theorem.
Definitions of partial and directional derivatives, differentiable functions,
gradient, Prop .: a continuous differentiable function and has all the directional derivatives.
Schwarz's Lemma total differential theorem. Functions
Ck, chain rule. Hessian matrix.
Taylor's formula at second order. Maximum and minimum stationary points
Positive definite matrices.
Prop: maximum or minimum points are critical points; the critical points in which the
Hessian matrix is positive (negative) are minimum (maximum) points; the points
critics in which the Hessian matrix has a positive and a negative eigenvalue are saddles.
Functions that can be differentiated from Rn to Rm; Jacobian matrix. Jacobian matrix of the
composition.
2. Normed spaces and Banach spaces
Examples. Converging and Cauchy sequences. Equivalent rules. Equivalence of the norms in Rn. The space of the
continuous functions with the sup norm a Banach space.
The fixed point theorem in Banach spaces Teo. 6:10
3. Implicit functions
The theorem of implicit and Inverse functions.
Constrained maxima and minima, Lagrange multipliers.
4. Ordinary differential equations
Examples: equations with separable variables, linear systems with constant coefficients
(solution with matrix exponential).
Existence and uniqueness theorem.
Linear systems, structure of solutions, wronskian, variation of constants.
Core Documentation
Analisi Matematica II, Giusti - Analisi Matematica II, ChierchiaType of delivery of the course
4 hours of frontal teaching 2 of exercises and 2 of tutoring a week In the event of an extension of the health emergency from COVID-19, all the provisions that regulate the methods of carrying out the teaching activities will be implemented. In particular, the following methods will apply: live remote lesson and recording of the lesson itself.Attendance
course attendance is strongly recommendedType of evaluation
written test and subsequent oral test