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Abstrak :
Misalkan U dan V adalah ruang vektor atas lapangan F. Hasil kali tensor (tensor product) dari U dan V adalah pasangan ruang vektor U_0 atas lapangan F dan pemetaan bilinier t:U×V→U_0, sedemikian sehingga (U_0,t) membentuk pasangan universal untuk bilinieritas (universal pair for bilinearity). Ruang vektor U_0 dinotasikan U⊗V. Yokonuma (1992) memberikan cara mengkonstruksi hasil kali tensor dari U dan V dengan menggunakan basis untuk U, V dan U_0. Roman (2008) memberikan cara lain untuk mengkonstruksi hasil kali tensor dari U dan V yaitu dengan menggunakan ruang hasil bagi F_(U×V)⁄S dan pemetaan bilinier t:U×V→F_(U×V)⁄S. Beberapa sifat yang dimiliki hasil kali tensor antara lain mengawetkan basis U dan V, anggota ruang vektor U_0 bersifat bilinier dan memiliki representasi tunggal, serta dapat membentuk isomorfisma antara himpunan pemetaan bilinier pada U×V dan himpunan transformasi linier pada U⊗V. Tugas akhir ini meninjau cara mengkonstruksi hasil kali tensor berdasarkan Yokonuma dan Roman, membahas beberapa sifatnya, dan memberikan contoh konstruksi hasil kali tensor dari R^2 dengan R^3 serta hasil kali tensor dari R^2 dengan M(2,2;R). ...... Let U and V are vector spaces over a field F. Tensor product of U and V is a pair of a vector space U_0 over a field F and a bilinear map t:U×V→U_0 such that (U_0,t) is a universal pair for bilinearity. The vector space U_0 is denoted by U⊗V. Yokonuma (1992) gave a way to construct a tensor product of U and V with bases of U, V and U_0. Roman (2008) gave a different way to construct a tensor product of U and V. It is constructed by using a quotient space F_(U×V)⁄S and a bilinear map t:U×V→F_(U×V)⁄S. Some properties of the tensor product are that it preserves the base of U and V, the elements of the vector space U_0 have a bilinear property and a unique representation. Furthermore, the tensor product can form an isomorphism between a set of bilinear map on U×V and a set of linear transformation on U⊗V. This skripsi gives two constructions of the tensor product based on Yokonuma and Roman, discusses some of its properties and gives examples of tensor product of R^2and R^3and tensor product of R^2 and M(2,2;R)

Abstrak :
This is the fifth and revised edition of a well-received textbook that aims at bridging the gap between the engineering course of tensor algebra on the one hand and the mathematical course of classical linear algebra on the other hand. In accordance with the contemporary way of scientific publication, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. As such, this new edition also discusses such modern topics of solid mechanics as electro- and magnetoelasticity. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this textbook addresses graduate students as well as scientists working in this field and in particular dealing with multi-physical problems. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.

Abstrak :
ABSTRAK
Pengukuran metode magnetotelluric di lapangan umumnya memiliki desain arah pengukuran Utara-Timur yang tidak searah arah penjalaran gelombang elektromagnetik di bidang batas struktur. Tensor impedansi pada Metode Magnetotellurik terdiri dari berbagai komponen penyusun salah satunya yaitu tensor rotasi. Secara teknis, pengukuran tersebut dapat dilakukan, namun dapat menyebabkan kesalahan interpretasi yang diakibatkan perubahan nilai impedansi yang berkorelasi dengan nilai resistivitas dibawah permukaan. Untuk mengatasi kesalahan interpretasi tersebut, desain pengukuran harus disesuaikan dengan arah struktur dan memperhatikan keadaan geologi. Rotasi merupakan salah satu solusi menghadapi hal tersebut, dapat dilakukan dengan beberapa cara antara lain metode rotasi fix-angle dan rotasi strike-angle.Penelitian ini difokuskan untuk menganalisis seberapa besar pengaruh rotasi terhadap model bawah permukaan bumi secara 2-D dan 3-D menggunakan pengamatan data sintetik dan data riil. Dari kedua metode tersebut, Rotasi Fix-angle memberikan hasil yang lebih baik dalam memetakan bawah permukaan.

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ABSTRACT
Magnetotelluric method field acquisition commonly have northing-easting field layout which is often not parallel to electromagnetic fields at structure boundary. Impedance tensor in magnetotelluric method consist of many components, that one?s rotation tensor. Technically, acquisiton using that field layout can be done using that configuration, but may causes misinterpretation in the impedance value which have corelation to subsurface resistivity. To excel those misinterpretation, field layout should be justified against the structure and observe the geological setting. Rotation is one of solution to solve the problem, there are two kinds of method in rotation that is fix angle rotation and strike angle rotation.This research focused on analyzing how rotation will affect subsurface model on 2-D and 3-D using synthetic and real data. From both rotation, fix angle rotation gives better result interpretating subsurface.

Abstrak :
Tensor dapat dipandang sebagai multidimensional array, yang merupakan perluasan konsep dari matriks, sehingga dapat dikonstruksi bentuk umum dari hasil kali tensor yang sifatnya hampir sama dengan sifat perkalian pada matriks. Pada skripsi ini disajikan teori-teori yang digunakan seperti polinomial homogen, monomial berderajat alpha, dan resultan 2 polinomial untuk menjadi landasan teori utama dalam memahami determinan tensor. Tujuan dari skripsi ini adalah memperlihatkan dan memahami sifat-sifat determinan tensor. ......The tensor can be viewed as a multidimensional array, which is an extension of the concept of the matrix, so that the general form of the tensor result can be constructed which is almost the same as the multiplication property of the matrix. This undergraduate thesis presents the theories used such as homogeneous polynomials, monomial degrees of alpha, and resultant 2 polynomials to form the basis of the main theory in understanding tensor determinants. The purpose of this paper is to show the properties of tensor determinants.

Abstrak :
Permeabilitas batuan merupakan parameter penting dalam meningkatkan drilling success ratio dan monitoring reservoir geotermal. Keberadaannya dikontrol oleh fracture akibat stress. Salah satu metode untuk menentukan keberadaan zona permeabel yang dikontrol oleh rekahan atau patahan adalah MEQ microearthquake . Identifikasi dan analisis karakteristik fracture dapat digunakan untuk mengoptimalkan produktivitas. Data gempa mikro tidak hanya memetakan sebaran zona permeabel berdasarkan sebaran hiposenternya, tetapi juga mampu mengkarakterisasi zona fracture berdasarkan analisis mekanisme fokal dan momen tensor. Dari data MEQ lapangan 'X' dengan memanfaatkan waveform lokal tiga komponen telah dilakukan inversi momen tensor. Hasil penelitian menunjukkan bahwa sebaran fracture yang mengontrol permeabilitas memiliki dominasi arah orientasi strike yakni NW-SE dan NE-SW. Hasil analisis momen tensor menunjukkan pada lapangan bagian Utara di elevasi sekitar 1 km bsl ke atas didominasi komponen implosif, berkaitan dengan pergerakan batuan secara konvergen yang dapat berdampak pada potensi penurunan permeabilitas batuan reservoir. Lapangan bagian Utara di elevasi sekitar 1 km bsl ke bawah menunjukkan komponen-komponen eksplosif, berkaitan dengan pergerakan batuan secara divergen yang mengindikasikan distribusi permeabilitas di lapangan Utara secara keseluruhan tergolong baik. Namun tetap ada potensi dan indikasi penurunan permeabilitas karena jika pergerakan konvergen hasil komponen implosif terus terjadi akibat ekstraksi massa fluida dan tidak diimbangi dengan suplai fluida ke reservoir, maka akan berpengaruh pada sifat fisik reservoir, termasuk penurunan permeabilitas. Selain itu, hasil penelitian ini juga menunjukkan bahwa permeabilitas di zona Selatan cukup besar. ...... Rock permeability is an important parameter in improving drilling success ratio and monitoring of geothermal reservoir. Its existence is controlled by fracture due to stress. Identification and analysis of fracture characteristics can be used to optimize the productivity. MEQ microearthquake is a method that can be used to determine the existing of permeable zones controlled by fractures or faults. MEQ data not only map the permeable zone distribution based on its hypocenter, but also characterize the fracture zones based on analysis of focal mechanism and moment tensor. Moment tensor inversion has done using MEQ data by utilizing three components of local waveform. The results of this study indicate that the distribution of fractures that control permeability has dominant strike orientation direction ie NW SE and NE SW. The results of moment tensor analysis show in the northern field at elevation of about 1 km bsl upward is dominated by implosive components, related to convergent rock movement which can impact on potential decrease of permeability of reservoir rock. The northern field at elevation of about 1 km bsl down show explosive components, related to diverging rock movement which indicates the distribution of permeability in the North field as a whole is quite good. However, there are potential and indication of a decrease in permeability because if convergent motion continues to occur due to fluid mass extraction and is not balanced with fluid supply to the reservoir, it will affect the physical properties of the reservoir, including the decrease in permeability. In addition, the results of this study also indicate that permeability in the South zone is considerable.

Abstrak :
Tensor yang dipandang sebagai multidimensional array adalah bentuk umum dari suatu matriks. Oleh karena itu, dapat dikonstruksi bentuk umum dari hasil kali matriks yang disebut sebagai hasil kali tensor. Tujuan dari tulisan ini adalah menjelaskan inversi kiri dan inversi kanan suatu tensor. Pada tulisan ini disajikan karakteristik eksistensi inversi kiri dan inversi kanan orde k dari suatu tensor. Disajikan pula hasil terkait keserupaan suatu tensor. ......Tensor, which is seemed as multidimensional array, is a general form of matrix. Therefore, tensor could be constructed into general form of matrix product which is called tensor product. The aim of this writing was to explain the right and left inversion of tensor. In this research, there were characteristics of right and left extension of orde k of tensor provided, in addition, there was also a result involved of the tensor similarity.

Abstrak :
Bringing together key researchers in disciplines ranging from visualization and image processing to applications in structural mechanics, fluid dynamics, elastography, and numerical mathematics, the workshop that generated this edited volume was the third in the successful Dagstuhl series. Its aim, reflected in the quality and relevance of the papers presented, was to foster collaboration and fresh lines of inquiry in the analysis and visualization of tensor fields, which offer a concise model for numerous physical phenomena. Despite their utility, there remains a dearth of methods for studying all but the simplest ones, a shortage the workshops aim to address.