jeo.;a

• kshï; fj,djg foaYk / ksnkaOk mx;s i|yd meïfkkak .

• fyd|ska ijka fokak .

• úIh wdYs%; fmd;a lshjkak .

• úIh wdYs%; .egˆ úi|kak .

1.  Scalar Fields.

2. Directional Derivative.

3. Systems of Coordinates.

4.  Curvilinear Coordinates.   

5. Integral along a curve.

6.  Green's Theorem in a plane.

7.  Surface Integrals and Surface area.

8.  Volume integrals.

9.   Stokes Theorem.

10.  The Divergence Theorem.

11. Applications of Vector Analysis in Classical Field Theory.

Course Content:

 Method of Assessment:• End of Semester One hour written examination 70%

• Mid Term Examination 10%

• Computer Practicals 20%

• Tutorials will be given.

Reference · Calculus by James Stewart

· Calculus and Analytic Geometry (Addison & Wesley)

Thomas, G. B. & Finney, R. L.

· Vector Calculus by R Guptha.

· Any other Calculus book.

w¾: oelaùu - ugsgus mDIaG

= ( x, y, z, t ) wosY fCIa;%h i|yd = c ksh;hla u.ska ,efnk mDIaGj,g ugsgus mDIaG hehs lshkq ,efns.

Definition – Level surfaces

For the scalar field = ( x, y, z, t ), the surfaces given by = c, are called level surfaces.

MAT 128 1.0 Mathematical Tools and Computer Practicals II

1.  wosY fCIa;% / 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.

Definition Suppose a scalar is defined associated with each point in some region of space, then it is called a scalar field.

Consider the scalar field = x2+ y2+ z2+ t2 .

When t = to , the level surfaces are given by

x2+ y2+ z2+t2o = c.

i.e. x2+ y2+ z2= k .

When k = c - t0 0, it represents a system of spheres.

k = c - t0 0 úg fuhska f.da,

moaO;shla ksrEmKh fjs.

Two dimensional situation

When the scalar field is defined on a plane, by considering a system of Oxy axis in the plane, we can represent the scalar field as = (x, y, t).

Then the equation (x,y,t0) = c represents the level curves.

Eg. = 2t( x2-4xy2) Then the level curves are 2 t0 ( x2 - 4 x y2 ) = c i.e x2 – 4 x y2 = k

Consider the points P=(x, y, z) in the region where scalar field is defined. Let Q= (x+x, y+ y, z+ z) be an arbitrary closed point in the same region.

Then of change of the scalar field when we move from P to Q in the straight path is

PQ)P()Q(

Directional Derivative (osYd jHq;amkakh&

, and called Difference Quotient.

So rate of change of the scalar field at P along the direction PQ is

PQ)P()Q(

lim0PQ

This limiting value is defined as the Directional derivative of at P in the direction of PQ.

Notation : If is the unit vector in the direction of PQ,

Y

X

O

Z

Q

P

x

y

z

PD

Note : Directional derivative is a function of time.

we denote the directional derivative of by

Here

PQ2 = (x + x –x)2 + (y + y –y)2 + (z + z –z)2

= x2 + y2 + z2

kδzjδyiδx PQ

αPQ.1.cosi .PQ

δxiPQ .Also δxαPQ.cos So

PQδx

αcosHence

PQδz

γcos nd aPQδy

βcos ysimilar wa

Q

P

x

y

z

PQ

z)y,(x,-δz)zδy,yδx,x

(lim

PQ)P()Q(

lim0PQ0PQ

cosα

δx

δz)zδy,y(x,-δz)zδy,yδx,x(lim

0PQ

cosβδy

δz)zy,(x,-δz)zδy,yx, (

γ

δz

z)y,(x,-δz)zy,δx,xcos

(

γcoszy,x,

βcoszy,x,

αcoszy,x,

z)(

y)(

x)(

γcosβcosαcoszy,x, kji.)(kz

jy

ix

).(zyx

zy,x,kji

So directional derivative of the scalar field at P in the direction PQ is ).()( PPD

E.g.

For the scalar field tz2yx 22

find the directional derivative when t = to

i ) at ( 1, 2, 1) towards the point ( 3, 2, 0).

ii) at ( 3, 1, 2) in the direction kj2ia

Solution tz2yxz

ky

jx

i 22

ot2.k)y2.(jx2.i

i ) ki2PQ

121

222 zyxoktjyixP,,

)(

ktji o242

kiktjiPDSo o 25

1242 .)(

51

24 )( ot

)( ot 25

52

ii). 213222 zyxoktjyix ,,)( P

ktji o226

kj2ia

kj2i66

and6a

ktjikji o226266 .)( PD

)( ot21066

)( ot 536

E.g. For the scalar field xzsin2zyx 22 find the directional derivative

i ) at ( 1, 2, 0) towards the point ( 3, 2, 1).

ii) at ( 0, 1, 2) in the direction kj2i2a

Solution kxzcosx2yjyz2ixzcosz2x2 2

xzsin2zyx 22

i) at ( 1, 2, 0), k6i2 5

ki2l

directional derivative is .525

10

ii) at ( 0, 1, 2), kj2i4 directional derivative is

3kj2i2.kj2i4 .

35

Note: If the directional derivative is positive then the scalar field increases as going from P to Q along the given direction.

If the directional derivative is negative then the scalar field decreases as going from P to Q along the given direction.

Properties of the Directional Derivative

).()( PPD

θP cos.)(

θP cos)(

. (p)θ anden ngle betweis the aHere

)P(

P

So in the perpendicular direction

to ;the value of the scalar

field does not change.

0)(;when)i( PD2

πθ

)(P

πθorθ owhen)ii(

)()(then PPD

So we get the maximum value of the directional derivative along the direction of the gradient and minimum in the opposite direction.

E.g. Find the direction such that directional derivative of the scalar field ztyzx 23

• does not change.

• is maximum.

at ( 3, 1, 0 )

Solution ktyjzix3 22

kt1j0i330.1.3 22

kt1i27 2

• .0.1.3

Scalar field would not change in any direction perpendicular to

Let kcjbian be such a direction. Then

okcjbia.kt1i27 2

oct1a27 2 and b is arbitrary.

27c

t1

a2

27c,t1a 2

k27jit1n 2

• .0.1.3

Directional derivative is maximum along the direction

kti 2127

For the scalar field , the directional derivative at P in the direction of l is gradient is .

P ysoS l osYdjg wosY fCIa;%fha osYd jHq;amkakh fjs.

Summery

.PP D

.PP D