Mata kuliah Vektor & Matriks (FT1006) ini adalah pengganti mata kuliah.
Matematika Lanjut (FT1003). Perubahan yang terjadi adalah berpindahnya
beberapa.
FT UNIVERSITAS SURABAYA
VEKTOR & MATRIKS
KATA PENGANTAR Diktat ini disusun berdasarkan kurikum baru di Fakultas Teknik yang efektif tahun 1996/1997. Mata kuliah Vektor & Matriks (FT1006) ini adalah pengganti mata kuliah Matematika Lanjut (FT1003). Perubahan yang terjadi adalah berpindahnya beberapa materi kuliah : • integral rangkap turun ke Kalkulus II (FT0002). • analisa variabel kompleks dihapus, khusus untuk jurusan Teknik Elektro materi ini berdiri sendiri sebagai mata kuliah Variabel Kompleks (FT1008). Pembagian bab dan urutan materi juga dibuat lebih kompak, disesuaikan dengan jumlah SKS yang baru, yaitu 3 SKS per minggu. Bab I sampai dengan bab III diajarkan paro semester pertama (sampai dengan UTS), berisikan materi dasar matriks dan vektor, serta pengertian derivatif vektor. Bab IV sampai dengan bab VI diberikan sesudah UTS, berisikan pengertian sistem koordinat dan masalah eigen, ditutup oleh kajian integral vektor. Tampak jelas bahwa mata kuliah ini melatih mahasiswa untuk lebih memahami perilaku ruang dan kejadian di dalam ruang. Permasalahan teknik pada umumnya juga kejadian di dalam ruang, misalnya medan elektromagnetik dan teori aliran (perpindahan massa dan panas). Diktat ini diharapkan dapat menunjang berjalannya perkuliahan Vektor & Matriks di Fakultas Teknik. Di samping itu, konsultasi dengan buku rujukan tetap dianjurkan. Adapun buku rujukan yang dapat dipakai : • E. Kreyzig
: Advanced Engineering Mathematics ; John Wiley & Sons, 1993
• Wylie & Barrett : Advanced Engineering Mathematics ; McGraw-Hill, 1982 • Schaum’s Outline Series : • Frank Ayres : Matrices • M.R. Spiegel : Vector Analysis Akhir kata mudah-mudahan diktat ini bermanfaat bagi kita semua, selamat belajar ! Februari 1997 Sugata Pikatan
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FT UNIVERSITAS SURABAYA
VEKTOR & MATRIKS
DAFTAR ISI Kata Pengantar........................................................................................................ i Daftar Isi ................................................................................................................ ii BAB I. MATRIKS................................................................................................. Pengertian matriks ............................................................................................ Determinan....................................................................................................... Operasi aljabar pada matriks ............................................................................. Penjumlahan............................................................................................... Perkalian.................................................................................................... Aplikasi matriks................................................................................................ Penyelesaian sistem persamaan................................................................... Kriptografi................................................................................................. Regresi linier.............................................................................................. SOAL...............................................................................................................
1 1 2 3 3 4 5 5 7 7 8
BAB II. ALJABAR VEKTOR............................................................................ Cara penulisan vektor ..................................................................................... Cara komponen........................................................................................ Cara vektor satuan ................................................................................... Cara matriks ............................................................................................ Operasi aljabar pada vektor............................................................................. Penjumlahan............................................................................................. Perkalian.................................................................................................. Perkalian rangkap .................................................................................... Aplikasi vektor pada analitika ruang................................................................ Vektor posisi ........................................................................................... Garis lurus ............................................................................................... Bidang datar ............................................................................................ Jarak dalam ruang .................................................................................... Perpotongan dalam ruang......................................................................... Sudut dalam ruang ................................................................................... SOAL.............................................................................................................
BAB III. DERIVATIF VEKTOR ....................................................................... Kurva dalam ruang ......................................................................................... Arah tangensial dan normal ...................................................................... Arah binormal .......................................................................................... Rumus Frenet .......................................................................................... Bidang lengkung............................................................................................. Gradien.................................................................................................... Aplikasi ................................................................................................... Divergensi ...................................................................................................... Curl ................................................................................................................ Operasi terhadap komposisi fungsi .................................................................. Jumlahan fungsi ....................................................................................... Perkalian fungsi........................................................................................
24 24 25 27 28 28 29 31 32 33 34 34 35
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FT UNIVERSITAS SURABAYA
VEKTOR & MATRIKS
Derivatif orde dua........................................................................................... Anti-derivatif vektor ....................................................................................... Vektor irotasional .................................................................................... Vektor solenoidal..................................................................................... Integral vektor terhadap skalar................................................................. SOAL............................................................................................................. BAB IV. SISTEM KOORDINAT....................................................................... Transformasi................................................................................................... Vektor basis............................................................................................. Transformasi vektor basis......................................................................... Transformasi koordinat ............................................................................ Transformasi komponen vektor................................................................ Diferensial dalam ruang................................................................................... Diferensial panjang................................................................................... Diferensial luas......................................................................................... Diferensial volume ................................................................................... Sistem koordinat ortogonal............................................................................. Sistem koordinat silinder.......................................................................... Sistem koordinat bola .............................................................................. Operator del (∇) ............................................................................................. Gradien.................................................................................................... Divergensi................................................................................................ Curl ......................................................................................................... Transformasi linier .......................................................................................... Transformasi geometri ............................................................................. Similaritas................................................................................................ SOAL............................................................................................................. BAB V. MASALAH EIGEN............................................................................... Vektor eigen dan nilai eigen............................................................................ Diagonalisasi .................................................................................................. Aplikasi .......................................................................................................... Analisa bentuk kuadratik.......................................................................... Penyelesaian persamaan diferensial simultan ............................................. Proses Markov......................................................................................... SOAL............................................................................................................. BAB VI. INTEGRAL VEKTOR ........................................................................ Integral vektor dengan integrator skalar .......................................................... Integral vektor dengan integrator vektor ......................................................... Integral garis............................................................................................ Integral permukaan .................................................................................. Integral terhadap derivatif vektor .................................................................... Integral terhadap gradien.......................................................................... Integral terhadap curl ............................................................................... Integral terhadap divergensi ..................................................................... SOAL............................................................................................................. iii
