I. To acquire technics and methods regarding inverse and implicit functions in R^n with applications to constrained problems.
II. To acquire a good knowledge of the concepts and methods in the classical integration theory on R^n, and, in particular, on curves and surfaces in R^3 with corresponding applications in Physics.
II. To acquire a good knowledge of the concepts and methods in the classical integration theory on R^n, and, in particular, on curves and surfaces in R^3 with corresponding applications in Physics.
Curriculum
scheda docente
materiale didattico
Gronwall's Lemma. Lipschitz dependence on initial data (Prop. 8.10 [Ch])
Applications of the implicit function theorem: Lagrange multipliers method, constrained maxima and minima. (Prop. 7.9 [Ch]) Inverse function theorem, local diffeomorphisms. [Ch]
Integration theory: 1. Riemann integral in $\mathbb{R}^n$
Review of the Riemann integral in one dimension ([G], Sec. 12.1). Rectangles in $\mathbb{R}^2$,
compactly supported functions, definition of
Riemann-integrable function in $\mathbb{R}^2$ (and thus $\mathbb{R}^n$).
Definition of measurable set ([G], Def. 12.3); a set is measurable if and only if its
boundary has measure zero ([G], Prop. 12.1). Sets normal with respect to the Cartesian axes.
A continuous function on a measurable set is integrable ([G], Thm. 12.1). Fubini's
reduction theorem ([G], Thm. 12.2).
Change of variables formula for integrals (proof outline). Polar,
cylindrical, and spherical coordinates. Examples: calculation of centroids and moments of inertia.
2. Curves, surfaces, flux, and the divergence theorem.
Review of the cross product. Examples of manifolds. Regular curves and regular surfaces.
Coordinate changes. Length of a curve. Definition of a regular surface ([G], Def. 15.4).
Tangent plane and unit normal vector. Surface area ([G], Def. 15.6).
Surface integrals. Flux of a vector
field through a surface. Examples. Statement of the divergence theorem. Proof of the Divergence Theorem (for normal domains in $\mathbb{R}^3$).
The Curl Theorem (proven for normal domains in $\mathbb{R}^2$ and $\mathbb{R}^3$).
3. Differential forms and work. ([G])
Differential 1-forms; integral of a differential 1-form (work
done by a vector field); closed and exact forms. A form is exact if and only if the integral
over any closed curve is zero. Example of a closed form that is not exact.
Simply connected sets. A closed form on a simply connected set is exact.
Star-shaped sets; a closed form on a star-shaped domain is exact.
Course supplements: 2-forms, pull-backs, and push-forwards for vector fields and forms.
[G] Giusti Analisi Matematica 2
Programma
Differential equations: Existence and uniqueness theorem, existence times. (Thm. 8.8 [Ch])Gronwall's Lemma. Lipschitz dependence on initial data (Prop. 8.10 [Ch])
Applications of the implicit function theorem: Lagrange multipliers method, constrained maxima and minima. (Prop. 7.9 [Ch]) Inverse function theorem, local diffeomorphisms. [Ch]
Integration theory: 1. Riemann integral in $\mathbb{R}^n$
Review of the Riemann integral in one dimension ([G], Sec. 12.1). Rectangles in $\mathbb{R}^2$,
compactly supported functions, definition of
Riemann-integrable function in $\mathbb{R}^2$ (and thus $\mathbb{R}^n$).
Definition of measurable set ([G], Def. 12.3); a set is measurable if and only if its
boundary has measure zero ([G], Prop. 12.1). Sets normal with respect to the Cartesian axes.
A continuous function on a measurable set is integrable ([G], Thm. 12.1). Fubini's
reduction theorem ([G], Thm. 12.2).
Change of variables formula for integrals (proof outline). Polar,
cylindrical, and spherical coordinates. Examples: calculation of centroids and moments of inertia.
2. Curves, surfaces, flux, and the divergence theorem.
Review of the cross product. Examples of manifolds. Regular curves and regular surfaces.
Coordinate changes. Length of a curve. Definition of a regular surface ([G], Def. 15.4).
Tangent plane and unit normal vector. Surface area ([G], Def. 15.6).
Surface integrals. Flux of a vector
field through a surface. Examples. Statement of the divergence theorem. Proof of the Divergence Theorem (for normal domains in $\mathbb{R}^3$).
The Curl Theorem (proven for normal domains in $\mathbb{R}^2$ and $\mathbb{R}^3$).
3. Differential forms and work. ([G])
Differential 1-forms; integral of a differential 1-form (work
done by a vector field); closed and exact forms. A form is exact if and only if the integral
over any closed curve is zero. Example of a closed form that is not exact.
Simply connected sets. A closed form on a simply connected set is exact.
Star-shaped sets; a closed form on a star-shaped domain is exact.
Course supplements: 2-forms, pull-backs, and push-forwards for vector fields and forms.
Testi Adottati
[Ch] Chierchia Analisi Matematica -n[G] Giusti Analisi Matematica 2
Bibliografia Di Riferimento
[MFS] Marcellini Fusco Sbordone Analisi Matematica 2 [BDG] Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Analisi matematica, McGraw Hill, Milano, 2011 (seconda edizione);Modalità Frequenza
following the lectures is highly reccommendedModalità Valutazione
The exam consists of a written test (which can be replaced by passing two mid-term tests) covering the course topics, and an oral exam.
scheda docente
materiale didattico
Gronwall's Lemma. Lipschitz dependence on initial data (Prop. 8.10 [Ch])
Applications of the implicit function theorem: Lagrange multipliers method, constrained maxima and minima. (Prop. 7.9 [Ch]) Inverse function theorem, local diffeomorphisms. [Ch]
Integration theory: 1. Riemann integral in $\mathbb{R}^n$
Review of the Riemann integral in one dimension ([G], Sec. 12.1). Rectangles in $\mathbb{R}^2$,
compactly supported functions, definition of
Riemann-integrable function in $\mathbb{R}^2$ (and thus $\mathbb{R}^n$).
Definition of measurable set ([G], Def. 12.3); a set is measurable if and only if its
boundary has measure zero ([G], Prop. 12.1). Sets normal with respect to the Cartesian axes.
A continuous function on a measurable set is integrable ([G], Thm. 12.1). Fubini's
reduction theorem ([G], Thm. 12.2).
Change of variables formula for integrals (proof outline). Polar,
cylindrical, and spherical coordinates. Examples: calculation of centroids and moments of inertia.
2. Curves, surfaces, flux, and the divergence theorem.
Review of the cross product. Examples of manifolds. Regular curves and regular surfaces.
Coordinate changes. Length of a curve. Definition of a regular surface ([G], Def. 15.4).
Tangent plane and unit normal vector. Surface area ([G], Def. 15.6).
Surface integrals. Flux of a vector
field through a surface. Examples. Statement of the divergence theorem. Proof of the Divergence Theorem (for normal domains in $\mathbb{R}^3$).
The Curl Theorem (proven for normal domains in $\mathbb{R}^2$ and $\mathbb{R}^3$).
3. Differential forms and work. ([G])
Differential 1-forms; integral of a differential 1-form (work
done by a vector field); closed and exact forms. A form is exact if and only if the integral
over any closed curve is zero. Example of a closed form that is not exact.
Simply connected sets. A closed form on a simply connected set is exact.
Star-shaped sets; a closed form on a star-shaped domain is exact.
Course supplements: 2-forms, pull-backs, and push-forwards for vector fields and forms.
[G] Giusti Analisi Matematica 2
Programma
Differential equations: Existence and uniqueness theorem, existence times. (Thm. 8.8 [Ch])Gronwall's Lemma. Lipschitz dependence on initial data (Prop. 8.10 [Ch])
Applications of the implicit function theorem: Lagrange multipliers method, constrained maxima and minima. (Prop. 7.9 [Ch]) Inverse function theorem, local diffeomorphisms. [Ch]
Integration theory: 1. Riemann integral in $\mathbb{R}^n$
Review of the Riemann integral in one dimension ([G], Sec. 12.1). Rectangles in $\mathbb{R}^2$,
compactly supported functions, definition of
Riemann-integrable function in $\mathbb{R}^2$ (and thus $\mathbb{R}^n$).
Definition of measurable set ([G], Def. 12.3); a set is measurable if and only if its
boundary has measure zero ([G], Prop. 12.1). Sets normal with respect to the Cartesian axes.
A continuous function on a measurable set is integrable ([G], Thm. 12.1). Fubini's
reduction theorem ([G], Thm. 12.2).
Change of variables formula for integrals (proof outline). Polar,
cylindrical, and spherical coordinates. Examples: calculation of centroids and moments of inertia.
2. Curves, surfaces, flux, and the divergence theorem.
Review of the cross product. Examples of manifolds. Regular curves and regular surfaces.
Coordinate changes. Length of a curve. Definition of a regular surface ([G], Def. 15.4).
Tangent plane and unit normal vector. Surface area ([G], Def. 15.6).
Surface integrals. Flux of a vector
field through a surface. Examples. Statement of the divergence theorem. Proof of the Divergence Theorem (for normal domains in $\mathbb{R}^3$).
The Curl Theorem (proven for normal domains in $\mathbb{R}^2$ and $\mathbb{R}^3$).
3. Differential forms and work. ([G])
Differential 1-forms; integral of a differential 1-form (work
done by a vector field); closed and exact forms. A form is exact if and only if the integral
over any closed curve is zero. Example of a closed form that is not exact.
Simply connected sets. A closed form on a simply connected set is exact.
Star-shaped sets; a closed form on a star-shaped domain is exact.
Course supplements: 2-forms, pull-backs, and push-forwards for vector fields and forms.
Testi Adottati
[Ch] Chierchia Analisi Matematica -n[G] Giusti Analisi Matematica 2
Bibliografia Di Riferimento
[MFS] Marcellini Fusco Sbordone Analisi Matematica 2 [BDG] Michiel Bertsch, Roberta Dal Passo, Lorenzo Giacomelli, Analisi matematica, McGraw Hill, Milano, 2011 (seconda edizione);Modalità Frequenza
following the lectures is highly reccommendedModalità Valutazione
The exam consists of a written test (which can be replaced by passing two mid-term tests) covering the course topics, and an oral exam.