1 Computer Applications IMAT Mathematical Tools and Computer Applications I
2 MAT 103 1.0 - Mathematical Tools andComputer Applications I Course Learning Objectives: At the end of the course, you will be able to find the products of vectors. obtain tangential planes and normals. solve equations involving complex numbers. formulate a problem occurring in the real world as a mathematical problem. use mathematical techniques to solve such problems. present the solutions for the given problem.
3 Vectors. Vector Geometry. Vector Calculus. Complex NumbersMAT Mathematical Tools and Computer Applications I The brief syllabus: Vectors. Vector Geometry. Scalar Product, Vector Product and Triple products. Lines in 3-D, Skew and parallel lines. Vector Equation of a plane. Vector Calculus. The Fundamental trid. Scalar and Vector Fields, Gradient of a scalar field, Divergence and Curl of a vector field. Complex Numbers De Moiver’s Theorem. nth root of Unity.
4 Method of Assessment: End of Semester One ( 01 ) hour written examination 80% Mid Semester examination % Computer Practicals. Tutorials and exercises will be assigned. References: Calculus and Analytic Geometry (Addison & Wesley) Thomas, G. B. & Finney, R. L. Calculus by James Stewart Vector Calculus by R Guptha. Any Calculus book. foaYk i|yd meïKSu jeo.;a fjS' meïKSfuS m%;sY;h u; wjika úNd.hg fmkS isàug wjir ,efnkq we;.
5 Vectors ( ffoYsl) udk folla iys;j i,lkq ,nk fN!;sl rdYshlg ffoYslhla hehs lshkq ,nhs' ksoiqka úia:dmkh' m%fjS.h' ;ajrKh' f¾Ld LKavhlo by; ,laIK we;s ksid ffoYsl f¾Ld LKav u.ska bosrsm;a flf¾' B A isg B olajd osYdjo AB osf.ka ffoYslfha úYd,;ajho ksrEmKh fjS' A
6 Vector Geometry (( ffoYsl ùch )ffoYsl foll úYd,;aj iudk yd osYd iudka;r fjS kuS tu ffoYsl iudk fjS' If two vectors are parallel and their magnitudes are same then they are equal. ffoYsl foll wdl,kh iudka;rdi% kshuhg wkqj isoq flf¾' Addition of two vectors is according to the parallelogram law. C B fuu ffoYsl w;r fldaKh fjS' OABC iudka;rdi%h iuSmQ¾K lrkak' OC f¾Ld LKavfhka ffoYslhkays iuSm%hqla;h ,efnS' O A
7 C B OAC ;%sfldaKhg fldaihska iQ;%h fhoSfuka O A fuS ksid OAC ;%sfldaKhg ihska iQ;%h fhoSfuka
8 10 ksoiqk R iuSm%hqla;fha úYd,;ah yd osYdj fidhkak' 60 o fhoSfuka 10
9 10 R 60 o 10 ;sria n,h iy iuSm%hqla;h w;r fldaKh kuS fuS ksid ;sria n,h iy iuSm%hqla;h w;r fldaKh 30o fjS'
10 a foYslh λ wosYfhka .=K lsrSu i,luq' fuys λ hkq ;d;a;aúl ixLHdjls'ffoYslhla wosYhlska .=K lsrSu' / Multiplication of a vector by a scalar. a foYslh λ wosYfhka .=K lsrSu i,luq' fuys λ hkq ;d;a;aúl ixLHdjls' λ a hkq wNsY+kH ffoYslh fjS' tkuS ^1& λ = 0 ; λ a ys osYdj a ys osYdju jk w;r ^2& λ > 0 ; λ a ys osYdj a ys osYdjg m%;súreoaO iy ^3& λ < 0 ;
11 ffoYsl foll wka;rh' / Difference of two vectors. B O A ffoYslh BA g iudk iy iudka;r fjS'
12 ffoYslhl m%lafIamKh' / Projection of a vector. OA u.ska ffoYslh ksrEmKh fjS' OB iy AB ,uSnl fjS' A OB u; ys m%lafIamKh OB jk w;r O OB osYdjg ys ixrplh f,i w¾: oelafjS B ;s;A .=Ks;h fyda wosY .=Ks;h' / Dot Product or Scalar Product. iy ffoYslh w;r fldaKh α kuS iy ys ;s;a .=Ks;h ' f,i bosrsm;a lrk w;r fjS' If the angle between the vectors and is α then the dot product of and is denoted by and
13 B O A ksoiqk 10 60 o 10 ;s;A .=Ks;fha .=K'/ Properties of the Scalar Product. tkuS ;s;A .=Ks;h kHdfoaYH fjS' Scalar Product is commutative. iy ffoYsl ,uSnl fjS kuS kuS fyda fyda wNsY+kH ffoYsl fyda tajd tlsfkl ,uSnl fjS'
14 Where is a unit vector with and .l;sr .=Ks;h fyda ffoYsl .=Ks;h' / Cross Product or Vector Product. iy ys l;sr .=Ks;h f,i bosrsm;a lrk w;r The cross product of and is defined by fuys hkq iy iuÕ iqr;a moaO;shla f,ig jQ tall ffoYslhls Where is a unit vector with and . igyk hkq wosYhla jk w;r hkq ffoYslhls'
15 B O A ksoiqk 10 60 o 10 l;sr .=Ks;fha .=K'/ Properties of the Cross Product. tkuS l;sr .=Ks;h m%;skHdfoaYH fjS' Cross Product is anti-commutative. iy ffoYsl iudka;r kuS kuS fyda fyda wNsY+kH ffoYsl fyda tajd tlsfkl iudka;r fjS'
16 / Resolving of vectors. ffoYsl úfNaokh' BOAB ;%sfldaKhg ihska iQ;%h fhoSfuka O A fuS ksid
17 úfYaI wjia:dj Special case,uSnl osYd follg úfNaokh‘ Resolving in perpendicular directions B fuúg O A fuS ksid
18 ;%sudk LKavdxl moaO;shla wkqnoaOfhkawith respect to a three dimensional system of coordinates
19 ffoYsl wdl,kfhA .=K' Properties of vector additionffoYsl wdl,kh kHdfoaYH fjS' ffoYsl wdl,kh ix>gH fjS'
20 ys idOkh m%:ufhka nj fmkajuq" fuúg iEu m%ldYkhlau wNsY+kH ffoYslh jk fyhska m%;sM,h i;H fjS" 1 2 fuúg iy yd osYd iudk iy fuS ksid m%;sM,h i;H fjS"
21 3 fuúg iy ys osYd m%;súreoO fjS" ;jo" fuS ksid m%;sM,h i;H fjS" tkuS
22 B O A = OAB ;%sfldaKfha j¾.M,h igyk ;, rEmhl j¾.M,h A iy tu ;,hg ,uSnl ffoYslh n kuS An g rEmfha ffoYsl j¾.M,h (Vector area ) hehs lshkq ,efnS" by; ;%sfldaKfha ffoYsl j¾.M,h
23 ;s;a .=Ks;h / l;sr .=Ks;h iuSnkaO m%;sM,hehs .ksuq
24 ys idOkh úfYaI wjia:dj' ffoYslh iy ffoYsl folgu ,uSnl hehs .ksuq' fuys hkq iy iuÕ iqr;a moaO;shla f,ig jQ tall ffoYslhls fuys hkq iy iuÕ iqr;a moaO;shla f,ig jQ tall ffoYslhls fuu iudka;rdi% iurEmS yd wkqrEm mdo w;r wkqmd;h a ksid
25 fojeks wjia:dj' ffoYslhg iudka;rj iy ,uSnlj ffoYslh úfNaokh lruq' ;jo Ys osYdjka iudk fyhska fuf,iu .;a úg fuys
26 m