MAT 103 1.0 Mathematical Tools. wosY lafIa;% ( Scalar fields) w¾: oelaùu wjldYfha huÞ m%foaYhl we;s iEu,laIHhla yd ix>Ü;j wosYhla w¾: olajd we;s úg thg.

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MAT Mathematical Tools

wosY lafIa;% ( Scalar fields) w¾: oelaùu wjldYfha huÞ m%foaYhl we;s iEu,laIHhla yd ix>Ü;j wosYhla w¾: olajd we;s úg thg wosY lafIa;%hla hehs lshkq,efnÞ. ksoiqka fuÞih u;,laIHhl oSma;sh >k jia;=jl tla tla,laIHfha meyeh Definition Suppose a scalar is defined associated with each point in some region of space, then it is called a scalar field.

, ,  … u.ska wosY fCIa;% ksrEmKh lrkq,efnS. Scalar fields are denoted by , ,  …. ldÜishdkq wlaI moaO;shlg idfmaCIj, With respect to a system of Cartesian coordinates,   ( x, y, z, t), where t means time. fuys t hkq ld,h fjS. w¾: olaùu hà wosY fCIa;%hla ld,fhka iajdh;a; kuS thg i;; wosY fCIa;%hla hehs lshkq,efnS. Definition Suppose a scalar is independent of time, then it is called a steady scalar field.

Gradient ( wkql%uKh & ys wkql%uKh hehs lshkq,efnS. wosY lafIa;%h i|yd, hkakg E.g. ys wkql%uKh fidhkak. th f,i fyda wkql% fyda f,i bosrsm;a krkq,nhs.

Then For the convenience we write the system Oxyz as Ox 1 x 2 x 3. i.e. take x = x 1, y = x 2, z = x 3. Also, take

Neglecting α, we get E.g. For the scalar field φ and Ψ show that solution

So

ffoYsl lafIa;% ( Vector fields ) Suppose a vector is defined associated with each point in some region of space, then it is called a vector field. E.g. Velocity of a moving fluid Magnetic field lines  for any scalar field . w¾: oelaùu wjldYfha huS m%foaYhl we;s iEu,laIHhla yd ix>Ü;j ffoYslhla w¾: olajd we;s úg thg ffoYsl lafIa;%hla hehs lshkq,efnS' Vector fields are denoted by A, B, C,….

We define the divergence of a vector field A as is, The curl of A is defined by where A =

In general we can define divergence and curl as Note Using these formulae you need to solve identities involving the gradient of a scalar field, divergence and curl of a vector field.

Eg. Solution : Show that Note

Eg. Solution : Show that

Similarly So

MAT Mathematical Tools

ffoYsl iólrK õp,H / ksh; ffoYsl iy wosY w¾:j;a whqrska ;s;a.=Ks;h yd / fyda l;sr.=Ks;h uÕska iuÞnkaOj we;s iólrKhla ffoYsl iólrKhla f,i ye|skafjÞ. Vector Equations An equation involving variable / constant vectors with dot product / cross product meaningfully is called a vector equation. ksoiqk ^1& fuhska f¾Ldjla ksrEmKh fjÞ. ^2& fuhska ;,hla ksrEmKh fjÞ. Solving of Vector Equations Type Iscalar non-zero vector when the solution is

Type IIscalarnon-zero vector and are perpendicular where is an arbitrary vector. Type III scalarnon-zero vectors

Type IV non-zero vectors so, and are perpendicular. since Type V non-zero vectors so it represents a straight line.