Introduction

• I believe that the step from MAT137 to MAT237 is easier than the step from high school to MAT137 because you already have some intuitions about the one variable case, but you should review each topic from MAT137 before we generalize it to the several variable case.
• The chapter on topology may be a little bit like the chapter on logic in MAT137: it may seem abstract at first but that's a new language that will be useful and powerful later in the course. Hold on: it will become more natural with practice!
• Contrary to MAT137, MAT237 is not an inverted class.
• There is no required textbook: the mandatory reading consists in the online notes.
  Nevertheless, if you want to go further, you can have a look at Folland's Advanced Calculus (be aware that it covers more material than MAT237). For a more computational point of view, you can have a look at Vector Calculus by Marsden and Tromba.
• In the first chapter (Preliminaries), we are going to review things that you already know. Don't worry, I'll go slower with the new material.

Preliminaries

Cartesian product and n-tuples

• Definitions and notations

Functions

• Definition and notation
• Image and preimage (or inverse image) of a set by a function
• Injective/surjective/bijective functions
• Inverse function
• Beware with the notation f ^-1(F) for the preimage: that's just a notation, the inverse function is not involved here, and the preimage is well defined for any functions (not only the bijective ones). But I agree that this notation is ambiguous.

Geometry of the Euclidean spaces

• ℝⁿ: definition, addition, scalar multiplication
• Dot product from the algebraic point of view: algebraic definition and algebraic properties

• Read the syllabus.
• Read section 0.1 Prerequisites of the online notes.
• Read the corresponding materials in the online notes.
• Study f₇ and f₈ from Slide 2.
• Work on slide 6.
• Prove the remaining properties of the dot product.

Preliminaries (continuation)

Geometry of the Euclidean spaces (continuation)

• Geometric definition of the dot product
• Algebraic definition of the cross product
• Properties of the cross product
• Geometric definition of the cross product

• Read section S0.2 of the online notes
• Prove the remaining properties of the cross product
• At the end of the lecture, we went very fastly on why both definitions of the cross-product are equivalent: convince yourself that they are indeed equivalent! Ask me if you have any question.

Preliminaries (continuation)

How to visualize a several variables function

• Definitions: level sets and graphs
• We went hiking in the massif de l'Esterel along the border between the Alpes-Maritimes and the Var.
• We went hiking along the French-Italian border from Sospel to Menton with a detour at the Monte Grammondo for a picnic.

Some topological notions

Balls and spheres

• Definitions: open balls, closed balls, spheres, bounded sets

Interior/Closure/Boundary of a set

• Definitions and first properties

• Read Chapter 0 of the online notes
• Play with the SAGE applets in the section 0.3 of the online notes or directly at SageMathCell.
  You can define a 2-variable function with:
  def f(x,y):
    return (x^2)/9 -(y^2)/4
  Then you can draw its graph with:
  plot3d(f,(-5,5),(-5,5), frame=True, axes=False, adaptive=True, color=rainbow(60, 'rgbtuple'))
  Or its level sets with:
  contour_plot(f, (-5,5), (-5,5), fill=False, contours=20, labels=True, label_inline=True)
  
• Work on the in-class questions about the interior/closure/boundary of a set.
• Work on the questions from the section 0.P.

Some topological notions (continuation)

Open and closed sets

• Definition: a set is open if it is equal to its interior.
• Characterization 1: a set is open if and only if it doesn't contain any of its boundary points (i.e. we don't need to remove anything to get the interior).
• Characterization 2: a set is open if and only if any of its points is the center of an open ball included in the set (i.e. starting from any point in the set, we can go in any direction without leaving the set).
  The above characterization is why we will use open sets in calculus.
• Definition: a set is closed if it is equal to its closure.
• Characterization 1: a set is closed if and only if it contains all of its boundary points (i.e. we don't need to add anything to get the closure).
• Theorem: a set is closed if and only if its complement is open.

Limits of multivariable functions

• We generalized the definition from real-valued functions defined on an interval seen in MAT237 to functions from a subset S of ℝⁿ and codomain ℝ^k. We noticed that we need to restrict to limit points of S for this definition to make sense: for instance, at an isolated point, we can find a delta small enough such that the implication is vacuously true.

• Do the in class questions about open and closed sets.
• Do the questions in Section 1.1 of the online notes.

• The schedule of the lectures is available in the introduction slides. You are expected to have reviewed the related notions from MAT137 before the class.
  For instance, it was easier to follow today's lecture if you had in mind the basic definitions/properties from MAT137 about limits/continuity for one-variable functions.
  Hence, be sure to review the basics about differentiability from MAT137 before the related lectures forthecoming in October (at least to understand well the definition seen in MAT137), that will be helpful.
• As I explained during the first lecture, we are going to use a lot the basic operations about functions and sets (for instance image and preimage of a set by a function, as in the characterization of continuity seen today).
  If you are not familiar with these notions, you should review the definitions from the first lecture and practice (I believe there are some exercises in last year MAT137 textbook, otherwise, ask me for exercises).
• Within a chapter, each lecture relies on the material from the previous lectures (for instance we first introduced open balls, which were used later to define the interior/closure/boundary of a set, which were used in turn to define open and closed sets, which were used today to characterize continuous functions).
  Hence, be sure that you understand the content of the previous lectures before each lecture.
• Be sure that you understand the examples seen in class! If you don't understand an example, ask me in class!
  For instance, today's examples about limits were chosen to convince you that we can't generalize the "compare the limit on the right and on the left" from MAT137 because we have a lot of more freedom to approach a point in ℝⁿ.
• Ask questions in class! Go to the office hours!
• Practice makes perfect: do the questions I give in class and, after the lecture, do the questions at the end of each section in the online notes.
• Understanding a new mathematical notion is a long-term process: it's totally normal to not understand right away. You shouldn't give up after only one attempt and it's totally fine to struggle a little bit first (mathematicians spend all their time struggling with new concepts). You should come back to the material of the lecture as many times as necessary. A direct corollary is that it is a very bad idea to start learning just before the test.

Some topological notions (continuation)

Limits of multivariable functions

• Limit points: definition, characterization and examples.
  Intuitively a limit point of S is simply a closure point of S which is not isolated in S.
• Limits: definition and properties.
  A vector-valued function admits a limit if and only if its components admit a limit, so it is enough to understand very well the real-valued case.
• Examples of computations: it is not enough to compute the limit along the lines (or any other curves) through the point.
• Very useful inequality: ∀(x,y)∈ℝ²\{(0,0)}, |xy|/(x^2+y^2)≤1/2

Continuity of multivariable functions

• Continuity at a point: ε-δ definition and characterization in terms of open balls.
  A vector-valued function is continuous at a point if and only if its components are continuous at this point, so it is enough to understand very well the real-valued case.
• The limit laws remain true: we can build continuous functions from the elementary functions.
• Topological characterization: a function is continuous if and only if the inverse image of an open set is open if and only if the inverse image of a closed set is closed.
  The proof is a little bit more difficult than the other proofs we studied, that's normal! No worries!

• Do the questions of Section 1.2 in the online notes.

Some topological notions (continuation)

Sequences

• Definitions: sequence, convergent sequence, subsequence, bounded sequence.
• Remark: a vector valued sequence is convergent if and only if its components are convergent.
  So it's enough to understand well the real-valued case.
• Very important lemma: if a sequence is convergent to L then each of its subsequences is also convergent to L.
  Beware: we found a non-convergent sequence admitting a convergent subsequence. So it's not enough to show that a subsequence is convergent to deduce that the original sequence is convergent.
• Theorem: if a convergent sequence takes values in a subset S then its limit is in the closure of S.
  Particularly, a sequence can't escape from a closed set, which is a very powerful property to check that a set is not closed.
• Theorem: Any bounded sequence admits a convergent subsequence.

Compactness

• Sequential definition: a subset S is compact if and only if any sequence with values in S admits a convergent subsequence whose limit is in S.
• Geometric definition: a subset S is compact if and only if it is closed and bounded.
• Theorem (EVT++): the continuous image of a compact set is compact.
• Theorem: a compact subset of ℝ admits a min and a max.
• Corollary (EVT): a continuous real-valued function defined on a compact set admits a min and a max.

Some topological notions (continuation)

The IVT: path-connected sets

• Definition
• Proposition: if two path-connected sets have a non-empty intersection then their union is path-connected.
  Beware, the non-empty intersection assumption is crucial here!
• Lemma: the path-connectedness subsets of ℝ are exactly the intervals.
• Theorem: the continuous image of a path-connected set is path-connected.
• Corollary: the generalized IVT

Differentiability

Real valued functions

• Directional derivatives: definition, geometric interpretation, examples, properties.
• Partial derivatives: definition, examples, "in practice computations".
  Recall that a partial derivative is a directional derivative so you can apply the properties of the directional derivatives.
• Gradient: definition.
  There is a small difference with the definition in the online notes but you shouldn't worry about it. In this section we define the gradient as the vector whose components are the partial derivatives whenever they exist (in the lecture notes, the gradient is only defined for differentiable functions).
• Differentiable functions: definition, geometric interpretation.

Differentiability (continuation)

Real valued functions (continuation)

• Theorem: if a function is differentiable at x then its differential at x is uniquely determined.
  The uniqueness of the differential relies on the openness of the domain: that's mainly why we assume that the domain is open.
• Theorem: if a function is differentiable at x then it is continuous at x.
• Proposition: differentiation rules.
• Theorem: if f is differentiable at x then (1) all its directional derivatives at x exist (2) and they can be computed in terms of the differential (3) moreover the differential is given by taking the dot-product with the gradient.
  The proof is postponed to Thursday.

Differentiability (continuation)

Real valued functions (continuation)

• Theorem: if f is differentiable at x then (1) all its directional derivatives at x exist (2) and they can be computed in terms of the differential (3) moreover the differential is given by taking the dot-product with the gradient.
• Proposition: if f is differentiable at x then the direction of the gradient is the direction of fastest increase and the magnitude of the gradient is the instantaneous rate of change in that direction.
• Theorem: if the partial derivatives of f exist in U and are continuous at x then f is differentiable at x.
• Have a look at the slides for a summary of this chapter.

Differentiability (continuation)

Linear maps and matrices: recollection

• Definitions and some properties

Differentiation of vector-valued functions

• Definition: differentiability and differential (or total derivative).
• Theorem: a vector-valued function is differentiable at a point if and only if its components are.
  In this case the differential is given by the differentials of the components.
• Proposition: if a vector-valued function is differentiable at a point then it is continuous at this point.
• Definition: Jacobian matrix.
• Theorem: if a vector valued function is differentiable then the directional derivatives of its components all exist and moreover the matrix of its differential is the Jacobian matrix whose entries are the partial derivatives of the components.
• Theorem: if all the partial derivatives of the components of a vector-valued function exist and are continuous at a point then the function is differentiable at this point.

Differentiability (continuation)

The Chain Rule

• Theorem: if f is differentiable at x₀ and if g is differentiable at f(x₀) then g∘f is differentiable at x₀ and its differential is given by composition of the differentials.
• Corollary: the Jacobian matrix of g∘f is given by the product of the Jacobian matrices of g and f.
• Corollary: the chain rule formula for the partial derivatives.

• There are as many notations as people: if you take two different textbooks/mathematicians randomly, there are 150% of chance (I might be exaggerating a little bit, but not that much!) that they are not using the same notation.
  For instance, I've already seen the following notations for the partial derivative of f with respect to the first variable (i.e. the directional derivative along e₁):
• The notations might be confusing at first: be sure that you understand what you are reading and/or writing! Use the context for that!
  For instance, given a function f:ℝ²→ℝ, simply denotes the derivative with respect to the first variable (i.e. the directional derivative along e₁), do not try to interpret the x in the denominator ∂x.
  Therefore, if you see it means that you FIRST compute the partial derivative and THEN that you evaluate it at (x²,xyz). You should NOT compute f(x²,xyz) and then take the derivative with respect to x.

Differentiability (continuation)

The chain rule (continuation)

• We worked on some examples
• We proved that the gradient is orthogonal to the level sets (after having properly defined what does it mean for a vector to be tangent to a level set).

The Mean Value Theorem

• Theorem: the MVT (Beware: you can only apply the MVT for two points such that the segment line between them is included in the domain).
• Definition: convex sets.
• Remark: if the domain is open and convex then we may apply the MVT to any couple of points in the domain. Particularly we obtained a MVT-like inequality in this case.
• Corollary: let f be a real-valued function whose domain is convex and open. If its gradient is always the zero vector then the function is constant.
• Corollary: let f be a real-valued function whose domain is path-connected and open. If its gradient is always the zero vector then the function is constant.

Differentiability (continuation)

Higher-order partial derivatives

• Definition/notation: higher-order partial derivatives.
• Beware: we've seen on an example that the order was very important! Taking the derivative with respect to x and then with respect to y may not give the same result as taking the derivative with respect to y and then with respect to x.
  As for the composition, we read the order from right to left.
• Clairaut's theorem: if a function is of class C² then the order is not important for the partial derivatives of order 2
• Corollary: if a function is of class C^k then the order is not important for the partial derivatives of order less than or equal to k.

Beware: I think that I made TWO mistakes today in class (It looks like that the reading week is necessary for me too...).

Differentiability (continuation)

Taylor's theorem: the one-variable case

• Taylor-Young's theorem
• Taylor-Lagrange's theorem

Taylor's theorem: the multi-variable case

• Taylor-Young's theorem at order 1: differentiability.
• Taylor-Lagrange's theorem at order 2.
• Taylor-Young's theorem at order 2.
• Definition: Hessian matrix.
• Taylor-Young's theorem at higher order (extra-curricular).
• Example: how to compute the Taylor polynomial of a multivariable function.
  Usually we don't use the definition of the Taylor polynomial because there are too many partial derivatives to compute. Instead, we rely on the one-variable Taylor's series that you already know.

Differentiability (continuation)

Critical points

• Definitions: critical points, local extrema.
• Theorem (first derivative test): a local extremum is a critical point.
• We saw why the Hessian matrix is going to be involved to study critical points of a C² function.
• Definitions about a symmetric matrix: definite positive/non-negative/negative/non-positive, degenerate, non-degenerate.
• Theorem: characterizations in terms of eigenvalues.
• Theorem: characterization of a definite positive/negative symmetric matrix in terms of an inequality.

Differentiability (continuation)

Critical points (continuation)

• Theorem: the second derivative test
• Critical points: the two-variable case

Reviews for Test 2

• Questions from section 2.7 of the lecture notes
• Questions 5, 6 and 10 of the review questions

Reviews

Differentiability (continuity)

Optimization with contraints: Lagrange multipliers

• Theorem: Lagrange multipliers.
  Statement and sketch of proof with the geometric idea.

Differentiability (continuity)

Optimization with contraints: Lagrange multipliers

• Example 1: the AM-GM inequality.
• Example 2: distance from a point to a line in the Euclidean space ℝ³.

Differentiability (continuation)

Lagrange multipliers

• Example 3: largest area for a triangle of fixed perimeter.
• Example 4: the spectral theorem.

The implicit function theorem

• Statement, geometric idea and heuristic.

• Example: Lagrange multipliers theorem.
• Example: implicit function theorem.
• Extra-curricular: a few words on submanifolds.

The IFT & the IFT

The Implicit Function Theorem (recollection)

• Statement of the theorem (recollection)
• Geometric interpretation
• Statement in the real-valued case (p=1)

The IFT & the IFT

The Inverse Function Theorem

• Statement: if the Jacobian of a C¹ function from an open set U⊂ℝⁿ to ℝⁿ is invertible at a, then f locally a C¹-diffeomorphism at a (i.e. up to shrinking the domain and the codomain to two open sets containing respectively a and f(a)).
• We proved that the Implicit Function Theorem and the Inverse Function Theorem are equivalent!
  Beware: it simply means that if one is true then the other one is also true and if one is false then the other one is also false.
  See the box below if you are interested in a proof of these theorems.

Singular points

• Case 1: curves in the plane

The IFT & the IFT (continuation)

Singular points (continuation)

• General case

The IFT & the IFT (continuation)

Singular points (continuation)

• Examples

Transformations

• Theorem: let f be a C¹ function from U⊂ℝⁿ open to ℝⁿ which is also injective then the Jacobian of f is everywhere invertible if and only f(U) is open and f is a C¹-diffeomorphism from U to f(U).

Integration

Uniform continuity

• Definition, geometric interpretation, examples.
• Heine-Cantor theorem: a function continuous on a compact set is uniformly continuous.

• In-class questions: they give you some results to (dis)prove that a function is uniformly continuous and contain a few interesting examples.
• Read the review sheet concerning the one-variable integral from MAT137/157/1XX.

Integration

Supremum and Infimum (recollection)

• Definitions
• ε-criterion of the sup/inf

Darboux's integral

• Partition of a segment/line rectangle
• Properties of partitions
• Upper/Lower Darboux sum of a bounded function on a segment line/rectangle
• Properties of the upper/lower Darboux sums
• Upper/Lower Darboux integrals of a bounded function (they always exist)
• Darboux integral of a bounded function (may not exists)
• The ε-criterion for integrability
• Basic properties of Darboux's integral

• Start solving the questions from sections 4.1 and 4.2
• Read the review sheet about the one-variable case (and solve the questions from it)
• Find counter-examples I asked at the end of the lecture for the properties of the integral (the integral of the product is not the product of the integral, the absolute value of a function is integrable doesn't imply that the function is integrable and the integral of the absolute value is not the absolute value of the integral...)

Integration (continuation)

Darboux's integral (continuation)

• Basic properties of Darboux's integral: proofs and counter-examples.
• A continuous function on a segment line or a rectangle is integrable: statement and proof.
• Zero content sets: definition and first properties.

Integration (continuation)

Zero content sets

• Basic properties of zero content sets.
• If the discontinuity set of a bounded function defined on a rectangle has zero content then the function is integrable.

Integration over a set which is not a rectangle

• Definition and basic properties.

Integration (continuation)

Iterated integrals

• Statement of "Fubini's theorem" and comments.
• Corollary when f_x is integrable.
• Corollary when f is continuous.
  Beware: we can't replace continuous by "the discontinuity set has zero content" in this statement!!!
• One example of application.

Integration (continuation)

Change of variables

• Heuristic.
• The one variable case: statement and proof.
• The multivariable case: statement, idea concerning the geometric proof, comments.
• Examples: polar coordinates, cylindrical coordinates, spherical coordinates.

Integration (continuation)

Functions of the form F(x)=∫f(x,y)dy

• Example: in general we can NOT swap integral and limit (I gave a counter-example).
• Theorem: if f is continuous then F is continuous (i.e. we can swap integral and limit).
  Beware, there are some technical assumptions on the domain!
• Theorem: if df/dx_i exists and is continuous w.r.t. to x and y then F is C¹ and we can swap integral and derivative.
  Beware, we ask df/dx to be continuous w.r.t. to x and y, it is stronger than asking that f_y(x)=f(x,y) is C¹ for any y.

• Read Improper integrals in one variable: reviews (updated notes from my MAT137 section last year).

Integration (continuation)

Improper integrals

• The theory! We will work on examples next Tuesday.
• BEWARE: contrary to the one-variable case, we only consider ABSOLUTE convergence for improper integrals in the multivariable case!!!!

Integration (continuation)

Improper integrals (continuation)

• Examples, examples, examples, examples, examples... and again examples...

Vector calculus

Line integrals

• Definition: here curve≝simple regular parametrized C¹ curves.
  Explanations and a few comments about orientation!
• Definition: line integral of a real-valued function along a curve.
• Theorem: invariance w.r.t. the parametrization.
• Definition: arclength of a curve.
• Theorem: characterization of the arclength in terms of a line integral.
  This result is quite useful since it shows that the arclength does not depend on the parametrization and since it allows to compute it using integration techniques.
• Definition: line integral of a vector field along a curve.
• Discussion about orientation: BEWARE, the line integral of a vector field depends on the orientation!!!
• Theorem: the Gradient Theorem or the FTC for line integrals.

Vector calculus (continuation)

Green's theorem

• A lot of definitions: regular region in ℝⁿ, simple regular piecewise-C¹ closed curve in ℝⁿ, regular region of ℝ² with a piecewise smooth boundary which is positively oriented...
• Green's theorem: if S is a regular region of ℝ² with a piecewise smooth boundary which is positively oriented and if F=(P,Q):U→ℝ² is a C¹ vector field whose domain contains S then ∫_∂SPdx+Qdy=∫∫_S∂Q/∂x+∂P/∂y.
• We talked about the proof for a regular region that can be decomposed in elementary regions.

Vector calculus (continuation)

Green's theorem: examples and applications!

• Compute a line integral thanks to an easier double integral.
• Compute an area thanks to a line integral (use (P,Q)=(0,x) or (-y,0) or (-y/2,x/2) for instance).
• A criterion for a vector field to be conservative (i.e. gradient of a potential).

Surface integrals

• Surface integrals for real-valued functions.
• Orientation..

Vector calculus (continuation)

Surface integrals (continuation)

• Interpretation of the cross-product in the surface integral.
• Definition of the surface integral of vector fields.
• Physics interpretation of the surface integral for vector fields.
• We computed one example of surface integral of a vector field (Beware: there was a typo in the question, I fixed it in my notes).

Vector calculus (continuation)

grad/∇, curl/∇×, div/∇·

• Definitions and notations
• Product rules
• Composition: the composition of two consecutive arrows in the diagram below for ℝ³ gives 0
  {functions}-grad->{vector fields}-curl->{vector fields}-div->{functions}

Conservative vector fields and Poincaré lemma

• Characterizations of conservative vector fields: a continuous vector field F is a gradient field F=∇f if and only if any line integral along a closed curve gives 0 if and only line integrals over oriented curves are path-independent.
  Then we say that f is a potential of F (or -f in Physics). This notion was already used, at least, in 1690 by Huygens in his "Discours de la cause de la pensanteur" but the first usage of the name "potential" is due to Green in 1828.
• Theorem: if F:U→ℝⁿ is a C¹ conservative vector field where U⊂ℝⁿ then ∂F_i/∂x_j=∂F_j/∂x_i for any i,j=1..n.
• Theorem: when U is star-shaped then the converse is true, i.e. if ∂F_i/∂x_j=∂F_j/∂x_i for any i,j=1..n then F is conservative.
  We only proved the result for n=2, but it generalizes for any n.
  Actually we can relax a little bit the assumption on the domain: it is enough to assume that U is contractile but this notion is not part of MAT237 and then the proof is more difficult.
  However, this result is false when U is not contractile, see the following example!
• Counter-example when the domain is not contractile: I gave an example of a non-conservative vector field defined on U=ℝ²\{0} such that ∂F₁/∂x₂=∂F₂/∂x₁, namely F(x,y)=(y/(x^2+y^2),-x/(x^2+y^2)).
  The history of this example is quite interesting: Clairaut proved a first version of Poincaré lemma in 1739 (here) but he didn't realize that he needed some assumptions on the domain. Then Jean Le Rond d'Alembert gave the counter-example seen in class in 1768 (here).
  The notation ∫_CP(x,y)dx+Q(x,y)dy was introduced by Clairaut in the above paper.

Vector calculus (continuation)

The divergence theorem

• Statement.
• A first example.
• Physics interpretation of the divergence.
• Gauss' law.
• How to use the divergence theorem to compute a volume using a surface integral.

Vector calculus (continuation)

Stokes' theorem

• The boundary in Stokes' theorem.
• Statement of Stokes' theorem.
• One example: how to compute a line integral using an easier surface integral.
• Physics interpretation of the curl.
• Another example: how to compute a surface integral using an easier surface integral.

Vector potentials

• Proposition: if a vector field is a curl then its divergence is 0.
• Theorem: if the domain is star-shaped then the converse is true and the proof of this formula gives a formula to compute a vector potential.
• Beware: we can relax the assumption on the domain (the domain needs to be contracitble) but the theorem is false in general with no assumption on the domain, you have already seen a counter-example with Gauss' law.