Medicine and Surgery HT

Introduction to the differential geometry module. Evaluation method: discussion. The derivative (or the differential in general) is like a "microscope" for analyzing curves and surfaces with the tools of linear algebra. Once entered into the microscopic linear world, the vectors and algebraic notions already tested for lines and planes (determinant, scalar product, vector product, rank, linear dependence, etc.) can thus also be applied to entities that are neither "rectilinear" nor planars". In this module, therefore, we will study some constants (eigenvalues, curvature, torsion, Gaussian curvature and Average curvature, etc.) which give important information on the shape and tactile perception of curves and surfaces. Their knowledge follows from notions of linear algebra grafted onto analysis calculations. The eigenvectors will often indicate privileged directions in which tactile perception reaches a maximum or minimum “pressure”. Meanwhile, let's see the eigenvectors in a different context but still inherent to curves: they simplify information that was initially encrypted (this already happened with linear applications). Rotation of an ellipse: the related polynomial equation loses the familiar canonical form; How to recognize an ellipse starting from a given second degree polynomial in two unknowns? We cannot pigeonhole the polynomial with our partial taxonomy. Artifice of the order 2 matrix relating to the second degree terms of the relevant polynomial. With this matrix it is possible to reconstruct the polynomial, through multiplication by appropriate vectors containing the variables x, y. The monomial in xy is the true disturbance term and its elimination occurs through diagonalization. Spectral theorem (orthogonal diagonalization) and orthogonal eigenvectors. Normalization of eigenvectors for rotation purposes (to avoid scaling changes). Coordinate change matrix and transpose matrix. The presence of orthogonal and normalized eigenvectors allows the transpose to be interpreted just like the inverse matrix, replacing the symbol (x,y) with the new symbol (X,Y) and activating the diagonalization process in order to eliminate the xy noise . Graphic representation of the ellipse in question, both in the rotated reference and in the original one. Transport of points from the new OXY reference to the old Oxy reference. Example of the focus of an ellipse. Transporting equations using the inverse law that allows you to replace the letters X, Y with functions of the letters x, y. Example of the director. Rotation of a parabola. Adjustment of the matrix by changing the direction (for a single eigenvector) or by exchanging the columns. Ellipse, parabola and hyperbola correspond to the positive, null, negative determinant. Matrix of order 3 and method of the invariant determinant for the canonical form of a parabola (can be extended without difficulty to the case of ellipse and hyperbola). Conics: general definitions and intermediate role of the parabola. New topic: curves. Initial example of an ellipse and a straight line, both crossed by a parametric moving point. Cartesian equation and parametric equations: the latter, placed inside the Cartesian equation, satisfy it. Curves in an Oxy reference. Example of the translated ellipse. Moving point expressed through the two component functions, x(u) and y(u). Generic tangent vector, P'(u). First derivatives of the two components and explicit calculation of the vector P'(u). Cartesian equation of the tangent line to a curve at a given point. New parameterization, using the variable s, with which we follow the curve at constant speed in modulus, unitary, and special tangent vector, t; t is therefore a unit vector and now its infinitesimal variation in direction faithfully informs us about the more or less curvilinear “elbow” path. Theorem, with proof: the derivative of t with respect to the variable s is a vector perpendicular to t. The derivative of the tangent unit vector t is equal to kn , where k (non-negative scalar) is the "curvature" and n is perpendicular to t in accordance with the theorem seen previously. Formula for calculating the curvature (to be demonstrated in the next lesson) for any parameterization, not necessarily the "rare" one, at unit speed (this last parameterization is the "curvilinear abscissa" and is often not available or is difficult to compute). Using this formula it is therefore possible to find a fundamental datum of uniform motion (the more or less "closed" shape of the curve section), even though only non-uniform motion is available. Curvature calculation examples. “Spiral” curve. Cardioid (Cartesian equation and parametric form). The “gradient” and its orthogonality with respect to the tangent vector (to be demonstrated in the next lesson). Gradient vector calculation examples. Demonstration of the formula for curvature. Fundamental ingredients: 1) interpretation of the function P(u) as composition P(s(u)) and consequent use of the derivation properties; 2) properties of the vector product (in fact, we immerse the curve in space and use the vector product as a filter to obtain the curvature information). During the demonstration we highlighted various details; in particular, P''(u) is not directed like the normal unit vector n, while P'(u) is directed like the tangent unit vector t. We also introduced the “binormal” unit vector b as a vector product of t and n. These three unit vectors constitute the so-called "Frenet trihedron", an apparatus that follows the curve at every point and records the curvature and other geometric properties of the curve (the "torsion", for example, as we will see). We have thus entered the geometry of space and will soon study examples of curves in space not contained in the Oxy plane. Curves in space. The curvilinear abscissa and other concepts inherent to curves in the Oxy plane are transported into this three-dimensional Oxyz context. Vectors t, n, b. Curvature (now in the numerator it is necessary to write the modulus of the vector product of P' and P''). Osculation plane and “torsion” τ. Formula for calculating torsion (without demonstration). Example of the elliptical helix. Surfaces in Oxyz space. Parameterizations with point dependent on two parameters: P(u,v). Cylinder example. Tangent vectors Pu and Pv. Curves contained in a given surface and composition with their respective curves in the two-dimensional domain. Calculation of the tangent vector of a curve through the vectors Pu and Pv. Tangent plane. Letters E, F, G. Torus example. Curves contained in the torus. OPIS questionnaires. Insights into curves and surfaces; further example in the bull, with a slightly more complex curve. Calculation of a tangent vector in the two known ways (passing through the torus, therefore using the tangent vectors Pu and Pv, or directly considering the curve immersed in space, without a host surface, therefore with the derivative in a single variable) . Area of a surface: this is a double integral which once again involves the letters E, F, G. The infinitesimal portion of the surface is linked to the vector product. Curvature of a surface (be careful, it is therefore not the curvature of a curve, which has already been studied). As the directions on the tangent plane of a given surface vary, we find curves with different curvatures and this variability gives, in short, an idea of the local shape of the surface ("under the microscope", at the point in question). Letters e, f, g. Gaussian curvature K. Elliptical, parabolic, hyperbolic points. Medium Curvature H. Formula for the two extreme curvatures. Cylinder case. Calculation of the extreme curvatures of a torus in 5 points, two elliptical, one parabolic and two hyperbolic. Alternative method: the extreme curvatures are the eigenvalues of the so-called "shape operator", a linear application in R2 which represents the reaction of the surface to the variation of our infinitesimal movement in any direction, on the tangent plane at a fixed point - topic to be revisited in the next lesson. Microscopic movements in the tangent plane and various consequences: fall in the same direction, opposite reaction of the surface, absence of reaction. Matrix of the shape operator. Summary of the proof that leads to the construction of the matrix. The columns of this matrix are the coordinates, in the basis {Pu , Pv} of the variation (derivative) of the normal vector N along the directions related to the variables u and v respectively. The extreme curvatures are then the eigenvalues of the found matrix. In many cases it is preferable to find the curvatures using the matrix rather than using the Gaussian curvature and the Mean (then solving the related quadratic equation). Conclusion, essentially, of the differential geometry module. Reflections and brief discussion on the affinities between research activity in mathematics and that in different fields, first of all medicine. Examples of research in mathematics. Graphs and their representation on a surface.

The literal calculation. Notable products. Decomposition of polynomials. Ruffini's theorem. Division between polynomials. Analogies between integers and polynomials. Irreducible polynomials. Algebraic equations. First degree inequalities. The Cartesian plan. The line. The conics (outline). Equations of the conics. The parable. Second level inequalities. Intersection between curves. Equation systems. Tangent line. Vectors and operations between vectors and on vectors (outline). Algebraic inequalities of a higher degree. Equations and rational inequalities. Irrational equations and inequalities. Overview of set theory. Overview of the logic of the propositions. Required, sufficient, necessary and sufficient conditions. Demonstration for absurd. The form of x. Equations and inequalities with the modules, also making use of graphic resolution. Power function with some graphs. Rules of powers. Exponential function. Exponential equations and inequalities. The logarithm and its basic properties. Graph of the logarithm. Logarithmic equations and inequalities. Overview of the inverse function. The trigonometric functions. The inverse trigonometric functions. Trigonometric equations and inequalities.

Notes of the course.

The lessons require compulsory attendance.