Yüksek Matematik II

İçerik

Dersin Amacı

Today, many branches of science—from operations research to statistics and economics—require the use of functions of several variables. Bilinear algebra is a fundamental tool in the analysis of these functions. Quadratic forms arise when one seeks an approximate representation of a multivariate function. In this context, determining whether a function has a minimum amounts to checking whether the quadratic form associated with the function is positive. Bilinear algebra also makes it possible to solve minimization problems by transforming them into problems of finding the shortest distance from a point to a set. Thus, when orthogonality is ensured, the minimum point is attained.

Dersin İçeriği

• Bilinear forms and inner product
• Pre-Hilbert spaces and Euclidean spaces
• Orthonormal bases for an inner product
• Orthogonal complement of a vector subspace
• Orthogonal projection theorem
• Applications: least squares, approximation of a periodic function
• Diagonalization of symmetric matrices
• Midterm exam
• Norms on a vector space, equivalence of norms in finite dimensions
• Continuity of a function of several variables
• Partial derivatives and the differential of a multivariable function
• Curves and surfaces: level curves, gradient vector, and tangent plane
• Minimum and maximum of a multivariable function
• Final exam

Dersin Öğrenme Çıktıları

At the end of this course, students will be able to:
1. Verify whether a given quadratic form defines an inner product and construct an orthonormal basis using the Gram–Schmidt process.
2. Determine the orthogonal complement of a vector subspace and use orthogonal projection to compute the distance between a vector and a subspace.
3. Diagonalize a symmetric matrix of dimension 3 or 4 by computing an orthonormal basis of eigenvectors.
4. Select an appropriate norm to study the continuity of functions of two or three variables and analyze the existence and continuity of their partial derivatives at a given point.
5. Compute the differential of a multivariable mapping.
6. Determine the gradient vector of a function, deduce a normal vector to level curves, and compute the tangent plane to an implicitly defined surface.
7. Determine local extrema of a function of two variables using the Hessian matrix.

Öğretim Yöntemleri

Cours et travaux dirigés.

Kaynaklar

Algèbre linéaire Joseph Grifone
Jean Marie Monier analyse 2

Konu Başlıkları

Değerlendirme Sistemi

Yeterlilik İlişkisi

AKTS - İş Yükü Tablosu