Transcription of A REVIEW ON VECTOR ALGEBRA & CALCULUS - GUC
1 ELECTROMAGNETIC PROF. REVIEW ON VECTOR ALGEBRA & CALCULUSEMFELECTROMAGNETIC PROF. quantity:has both magnitude and direction as velocity, force, electricfield1-Vectors and scalarsAA 1a zzyyxxaAaAaAaAaAA || 222||zyxAAAAA || AAa &Scalar quantity:has magnitude only as temperature, mass, volume or energyaAA ELECTROMAGNETIC PROF.
2 VECTOR :zyxa za ya xr zyxa za ya xr Let the position of source (S) is defined by:and the position of observation point P is defined by: rrR Thus, the distance (R) between source and observation point is:222)()()(zzyyxx RRa R xzySr r rrR )z ,y ,x(S )z ,y ,x(PRa R zyxa )(a )(a )(zzyyxx |||| rrRR ELECTROMAGNETIC PROF. algebraVector addition and subtraction:The following two vectors:zzyyxxa Aa Aa A A BA zzyyxxa Ba Ba B B &-Addition: zzzyyyxxxa B Aa B Aa B A BA A B -Subtraction: zzzyyyxxxa B Aa B Aa B A BA BA B A ELECTROMAGNETIC PROF.
3 Multiplication:-Scalar (dot) product: c o s || || . BABA The scalar (dot) product between two vectors is defined as the product of the magnitudes of , and the cosineof angle between them. The result is scalar A BA &B For zzyyxxa Aa Aa A A zzyyxxa Ba Ba B B &zzyyxxB AB AB A B . A A B c o sAELECTROMAGNETIC PROF. (cross) product:n s in || || BABA zyxzyxzyxB B B A A A a a a B A The VECTOR (cross) product between two vectors is defined as the product of the magnitudes of , and the sineof angle between them.
4 The result is VECTOR (which is perpendicular to the plane of ) A B&A B B&A A B n ya xa za +++___Twice the area of triangle between the two BA A ELECTROMAGNETIC PROF. C C B B B A A A )CB( . A A. Triple scalar product:, a Aa Aa A Azzyyxx zzyyxxa Ba Ba B B zzyyxxa Ca Ca C C a nd ) B A( . C) A C( . B) C B( . A :) C B( AC )B A( )C . B(A)C . A(BC )B A( Triple product of vectors:)B.
5 A(C-)C . A(B) C B( A A C n B A (B C) = ABC sin cos The volume of the parallelepiped A (B C) = B(A C) C(A B)This expression is known as the BACK-CAB PROF. systems: An orthogonal system is one in which the coordinates are mutually perpendicular. The most standard and commonly used are: Cartesian (or rectangular) Circular (cylindrical) SphericalThe solution of a given practical problem can be greatly facilitated by the proper choice of a coordinate system that best fits the geometry of the problemELECTROMAGNETIC PROF.)
6 Coordinates (x, y, z):ELECTROMAGNETIC PROF. d y d zSd yya d xd zSd zza d xd ySd Position VECTOR :zyxa a a zyxAAAA Differential length:zzyyxxa da da d d Differential area:xa d y d zSdx ya d z d xSdy za d y d xSdz Differential volume:d x d y d zd ELECTROMAGNETIC PROF. 2-Cylindrical coordinates ( , , z):ELECTROMAGNETIC PROF.
7 VECTOR :za a a zAAAA Differential length:za a a dzddd Differential area: a dzdSd a dzdSd za ddSdz Differential volume:dzddd a dzd Sd a dzdSd zza dd Sd d zddd d d d d dzdzELECTROMAGNETIC PROF. r3. Spherical coordinates (r , , ): 2000 rELECTROMAGNETIC PROF. VECTOR : a a a AAAArr Differential length: a s ina a drrddrdr Differential area:rrddrSda s in 2 a s in d r drSd a d r drSd Differential volume: dd r drd s in 2 d s in r dd rd sinrd2 d rrdrELECTROMAGNETIC PROF.
8 Between coordinate systemxa ya a a xy c o sx s in yzz 22yx xytan1- cos sin 0-sin cos ya za a a za Example: a s ina c o s a x yxa s ina c o s a 1-Cartesian Cylindrical : zz ELECTROMAGNETIC PROF. Cylindrical : a za ra a z r inrs osrzc 22zr ztan1- sin cos 0001cos -sin a a a a za Example:za c o sa s in a r a c o sa s in a rELECTROMAGNETIC PROF.
9 Sphericalza c o sa s in a r sin cos cos cos -Sin sin sin cos sin Cos cos -sin a a xa ya za zxsaa c o s)a in ( c o s s in y zxsaosa c o sa ins in c s in y 22222zyxzr ztanztan221-1-yx xytan1- osinrc s x inrys s in c o s rz za s ina c o s a zxsaa s in )a in ( c o s c o s y zxsaosa ina s in c o s c c o s y yxa c o sa s in a Example:ELECTROMAGNETIC PROF. PROF.
10 Line integral: (integration on path) zzyyxxdAdAdAd .A A Let zzyyxxaAaAaA A zzyyxxadadad d c d .A Surface integral: ssdSnAS .d .A sSd .A SA n (Flux of VECTOR field )A (Total outward Flux) Volume integral: vdv A cSv Circulation (rotation) of around CA dPathP1P24- VECTOR calculusELECTROMAGNETIC PROF. Differential operator (Del operator):zyxazayax It does not have any physical meaning, but once applied to a scalar physical quantity it attains a meaning, be definition.
