https://repositorio.ufms.br/handle/123456789/11021 2026-04-05T22:09:06Z https://repositorio.ufms.br/handle/123456789/14110 Título: Teorema Decomposição Espectral e suas aplicações Abstract: This work presents a structured study on the diagonalization of linear operators in finite-dimensional vector spaces. Initially, fundamental concepts of vector spaces, bases, dimension, and linear transformations are reviewed, with an emphasis on the Rank-Nullity Theorem (Teorema do Núcleo e da Imagem). The investigation then delves into the geometry of spaces equipped with an inner product, both real (Euclidean) and complex (Unitary), establishing the definitions of orthogonality, adjunction, and normal operators. The core focus of this monograph is the Spectral Theorem for Self-Adjoint Operators. It is rigorously demonstrated that such operators possess purely real eigenvalues and that eigenvectors corresponding to distinct eigenvalues are orthogonal, guaranteeing the existence of an orthonormal basis that diagonalizes the operator. The theory is illustrated through practical examples, including the analysis of symmetric matrices and differential operators in polynomial spaces, highlighting the elegance and applicability of spectral decomposition. Tipo: Trabalho de Conclusão de Curso 2025-01-01T00:00:00Z https://repositorio.ufms.br/handle/123456789/14108 Título: Integração Complexa: aplicações do Teorema dos Resíduos Abstract: This undergraduate thesis presents an in-depth study of Complex Analysis, emphasizing integration theory and the Residue Theorem. Initially, the necessary theoretical foundations are established, addressing the properties of complex numbers, holomorphic functions, Cauchy-Riemann conditions, and function representation via Power and Laurent series. The study proceeds to integration in the complex plane, detailing core results such as the Cauchy-Goursat Theorem and the Cauchy Integral Formula. The primary focus of the work lies in the formulation, proof, and application of the Residue Theorem, exploring techniques for calculating residues at isolated singularities. Finally, the effectiveness of this mathematical tool is demonstrated in solving improper real integrals and in interdisciplinary applications within Physics and Engineering, highlighting how complex analysis simplifies problems that are difficult to resolve using traditional real calculus methods. Tipo: Trabalho de Conclusão de Curso 2025-01-01T00:00:00Z https://repositorio.ufms.br/handle/123456789/11022 Título: Aproximação geométrica de áreas sob curvas utilizando o GeoGebra: Uma proposta envolvendo funções para o Ensino Médio Abstract: This work proposes an innovative approach to teaching the calculation of areas under curves in high school without directly resorting to integral theory. The proposal uses geometric approximation methods, such as rectangles and trapezoids, along with technological resources supported by the GeoGebra software to obtain approximations of the area below the curve y=f(x). The present proposal introduces the functions f(x)=x, f(x)=x², and f(x)=x³, defined over an interval [a,b], where the area calculation is performed through approximation by dividing the interval into n subintervals. This allows students to visualize and discuss the potential errors involved. The methodology combines practical manual calculation activities with GeoGebra, providing a comparative analysis of approximation methods. Students are encouraged to reflect on the accuracy of the approximations, recognizing through this approach that the greater the number of divisions into rectangles and trapezoids of the area to be calculated, the closer the estimated area will be to the true value as calculated by technological tools. This work not only develops mathematical reasoning but also promotes the use of technologies in teaching mathematics, making the learning process more dynamic and engaging. The conclusion highlights the importance of geometric approximations in area calculation, introducing students to the study of Riemann sums and integrals in advanced stages of their education. Tipo: Trabalho de Conclusão de Curso 2025-01-01T00:00:00Z