UNIT-II VECTOR CALCULUS Directional derivative The derivative of a point function (scalar or vector) in a particular direction is called its directional derivative along the direction. The directional derivative of a scalar point function φ in a given direction is the rate of change of φ in the direction. It is given by the component of grad φ in that direction. The directional derivative of a scalar point function φ (x,y,z) in the direction of → a is given by → → ∇ a a . φ . Directional derivative of φ is maximum in the direction of φ ∇ . Hence the maximum directional derivative is φ φ grad or ∇ Unit normal vector to the surface If φ (x, y, z) be a scalar function, then φ (x, y, z) = c represents A surface and the unit normal vector to the surface φ is given by φ φ ∇ ∇ Equation of the tangent plane and normal to the surface Suppose → a is the position vector of the point ) , , ( 0 0 0 z y x On the surface φ (x, y, z) = c. If → → → → + + = k z j y i x r is the position vector of any point (x,y,z) on the tangent plane to the surface at → a , then the equation of the tangent plane to the surface φ at a given point → a on it is given by 0 . = - → → φ grad a r If → r is the position vector of any point on the normal to the surface at the point → a on it. The vector equation of the normal at a given point → a on the surface φ is 0 = × - → → φ grad a r The Cartesian form of the normal at ) , , ( 0 0 0 z y x on the surface φ (x,y,z) = c is z z z y y y x x x o ∂ ∂ - = ∂ ∂ - = ∂ ∂ - φ φ φ 0 0 Divergence of a vector If ) , , ( z y x F → is a continuously differentiable vector point function in a given region of space, then the divergences of → F is defined by z F k y F j x F i F div F ∂ ∂ + ∂ ∂ + ∂ ∂ = = ∇ → → → → → → → → .
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UNIT-II VECTOR CALCULUS fileUNIT-II VECTOR CALCULUS Directional derivative The derivative of a point function (scalar or vector) in a particular direction is called its directional
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UNIT-II
VECTOR CALCULUS
Directional derivative
The derivative of a point function (scalar or vector) in a particular
direction is called its directional derivative along the direction.
The directional derivative of a scalar point function φ in a given
direction is the rate of change of φ in the direction. It is given by the
component of gradφ in that direction.
The directional derivative of a scalar point function
φ (x,y,z) in the direction of →a is given by
→
→∇
a
a.φ.
Directional derivative of φ is maximum in the direction of φ∇ .
Hence the maximum directional derivative is φφ grador∇
Unit normal vector to the surface
If φ (x, y, z) be a scalar function, then φ (x, y, z) = c represents
A surface and the unit normal vector to the surface φ is given by
φφ
∇∇
Equation of the tangent plane and normal to the surface
Suppose →a is the position vector of the point ),,( 000 zyx
On the surface φ (x, y, z) = c. If →→→→
++= kzjyixr is the position vector of
any point (x,y,z) on the tangent plane to the surface at→a , then the
equation of the tangent plane to the surface φ at a given point →a on it is
given by 0. =
−→→
φgradar
If →r is the position vector of any point on the normal to the surface
at the point →a on it. The vector equation of the normal at a given point
→a on the surface φ is 0=×
−→→
φgradar
The Cartesian form of the normal at ),,( 000 zyx on the surface
φ (x,y,z) = c is
z
zz
y
yy
x
xx o
∂∂−=
∂∂−=
∂∂−
φφφ00
Divergence of a vector
If ),,( zyxF→
is a continuously differentiable vector point function in
a given region of space, then the divergences of →F is defined by
z
Fk
y
Fj
x
FiFdivF
∂∂+
∂∂+
∂∂==∇
→→
→→
→→→→
.
=x
Fi
∂∂
→→
∑
If →→→→
++= kFjFiFF 321 ,then ).( 321
→→→→++∇= kFjFiFFdiv
i.e., z
F
y
F
x
FFdiv
∂∂+
∂∂+
∂∂=
→321
Solenoidal Vector
A vector →F is said to be solenoidal if 0=
→Fdiv (ie) 0. =∇
→F
Curl of vector function
If ),,( zyxF→
is a differentiable vector point function defined at each
point (x, y, z), then the curl of →F is defined by
→→
×∇= FFcurl
= z
Fk
y
Fj
x
Fi
∂∂×+
∂∂×+
∂∂×
→→
→→
→→
= x
Fi
∂∂×
→→
∑
If →→→→
++= kFjFiFF 321 ,then )( 321
→→→→++×∇= kFjFiFFcurl
321 FFF
zyx
kji
Fcurl∂∂
∂∂
∂∂=
→
=
∂∂−
∂∂+
∂∂−
∂∂−
∂∂−
∂∂ →→→
y
F
x
Fk
z
F
x
Fj
z
F
y
Fi 121323
Curl→F is also said to be rotation
→F
Irrotational Vector
A vector →F is called irrotational if Curl 0=
→F
(ie) if 0=×∇→F
Scalar Potential
If →F is an irrotational vector, then there exists a scalar function φ
Such that φ∇=→F . Such a scalar function is called scalar potential of
→F
Properties of Gradient
1. If f and g are two scalar point function that ( ) gfgf ∇±∇=±∇ (or)
( ) gradggradfgfgrad ±=±
Solution: ( ) ( )gfz
ky
jx
igf ±
∂∂+
∂∂+
∂∂=±∇
→→→
= ( ) ( ) ( )
±∂∂+±
∂∂+±
∂∂ →→→
gfz
kgfy
jgfx
i
= z
gk
z
fk
y
gj
y
fj
x
gi
x
fi
∂∂±
∂∂+
∂∂±
∂∂+
∂∂±
∂∂ →→→→→→
=
∂∂+
∂∂+
∂∂±
∂∂+
∂∂+
∂∂ →→→→→→
z
gk
y
gj
x
gi
z
fk
y
fj
x
fi
= gf ∇±∇
2. If f and g are two scalar point functions then ( ) fggffg ∇+∇=∇ (or)
ggradffgradgfggrad +=)(
Solution: ( ) =∇ fg ( )fgz
ky
jx
i
∂∂+
∂∂+
∂∂ →→→
= ( ) ( ) ( )
∂∂+
∂∂+
∂∂ →→→
fgz
kfgy
jfgx
i
=
∂∂+
∂∂+
∂∂+
∂∂+
∂∂+
∂∂ →→→
z
fg
z
gfk
y
fg
y
gfj
x
fg
x
gfi
=
∂∂+
∂∂+
∂∂+
∂∂+
∂∂+
∂∂ →→→→→→
z
fk
y
fj
x
fig
z
gk
y
gj
x
gif
= fggf ∇+∇
3. If f and g are two scalar point function then 2g
gffg
g
f ∇−∇=
∇ where
0≠g
Solution: =
∇g
f
∂∂+
∂∂+
∂∂ →→→
g
f
zk
yj
xi
= ∑
∂∂→
g
f
xi
= ∑
∂∂−
∂∂
→
2g
x
gf
x
fg
i
=
∂∂−
∂∂
∑ ∑→→
x
gif
x
fig
g 21
= [ ]gffgg
∇−∇2
1
4. If →→→→
++= kzjyixr such that rr =→
,prove that →
−=∇ rnrr nn 2
Solution: nn r
zk
yj
xir
∂∂+
∂∂+
∂∂=∇
→→→
=
∂∂+
∂∂+
∂∂ →→→
z
rk
y
rj
x
ri
nnn
= z
rnrk
y
rnrj
x
rnri nnn
∂∂+
∂∂+
∂∂ −
→−
→−
→111
=
++→→→
−
r
zk
r
yj
r
xinr n 1
=
++→→→−
kzjyixr
nr n 1
= →−
rr
nr n 1
5. Find a unit normal to the surface 422 =+ xzyx at (2,-2, 3)
Solution: Given that xzyx 22 +=φ
)2( 2 xzyxz
ky
jx
i +
∂∂+
∂∂+
∂∂=∇
→→→φ
= ( ) ( ) ( )xkxjzxyi 222 2→→→
+++
At (2,-2, 3)
( ) )4()4(68→→→
+++−=∇ kjiφ
= →→→
++− kji 442
63616164 ==++=∇φ
Unit normal to the given surface at (2,-2,3)
φφ
∇∇
=6
442
→→→
++− kji
=
++−→→→kji 22
3
1
6. Find the directional derivative of xyzxzyzx ++= 22 4φ at (1,2,3) in the