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VECTORCALCULUS
Prepared byEngr. Mark Angelo C. Purio
Differential Length, Area, and Volume
Differential Length, Area, and Volume
Differential elements in length, area, andvolume are useful in vector calculus.
They are defined in the Cartesian,Cylindrical, and Spherical coordinate
systems
Differential Length, Area, and Volume
A. Cartesian Coordinates1. Differential Displacement
2. Differential Normal Area
3. Differential Volume
Differential Length, Area, and Volume
Differential Elements in the Right-handedCartesian Coordinate System
Differential Length, Area, and Volume
Differential Normal Areas inCartesian Coordinates
Differential Length, Area, and Volume
A. Cylindrical Coordinates1. Differential Displacement
2. Differential Normal Area
3. Differential Volume
Differential Length, Area, and Volume
Differential Elements inCylindrical Coordinates
Differential Length, Area, and Volume
Differential Normal Areas inCylindrical Coordinates
Differential Length, Area, and Volume
A. Spherical Coordinates1. Differential Displacement
2. Differential Normal Area
3. Differential Volume
Differential Length, Area, and Volume
Differential Elements inSpherical Coordinates
Differential Length, Area, and Volume
Differential Normal Areas inSpherical Coordinates
EXAM
PLE 3
.1 Consider the objectshown. Calculate:a) The distance BCb) The distance CDc) The surface area
ABCDd) The surface area ABOe) The surface area
AOFDf) The volume ABDCFO
a) 10b) 2.5 πc) 25 π
d) 6.25 πe) 50f) 62.5 π
EXER
CISE
3.1 Disregard the differential
lengths and imagine that theobject is part of a sphericalshell. It may be describe as3 ≤ ≤ 5, 60° ≤ ≤ 90°,45° ≤ ∅ ≤ 60°where surface= 3 is the same as ,
surface = 60° is , andsurface ∅ = 45° is .
Calculatea) The distance DHb) The distance FGc) The surface area AEHDd) The surface area ABDCe) The volume of the
object
a) 0.7854b) 2.618c) 1.179
d) 4.189e) 4.276
Line, Surface, and Volume Integrals
Line, Surface, and Volume Integrals
By a line we mean the path along a curvein space.
Line, curve, and contour can be usedinterchangeably.
The line integral ∫ • is the of thetangential component of A along curve
L.
Line, Surface, and Volume Integrals
Given a vector field A and acurve L, we define the
integral
as the line integral ofA around L.
Path of integration of avector field
Line, Surface, and Volume Integrals
If the path of integration is a closed curve suchas abcba
Becomes a closed contour integral
which is called circulation of A around L
Line, Surface, and Volume IntegralsGiven a vector fieldA, continuous in aregion containing
the smooth surfaceS, we define the
surface integral orthe flux of Athrough S as The flux of a vector field A
through surface S
Line, Surface, and Volume Integralswhere at any pointon S , is the unit
normal to S.For a closed
surface (defining avolume):
The flux of a vector field Athrough surface S
Which is referred to a thenet outward flux of A from
S.
Line, Surface, and Volume IntegralsWe define
as the volume integral of the scalar overthe volume .
The physical meaning of the line, surface, orvolume integral depends on the nature of the
physicsl quantity represented by A or .
EXAM
PLE 3
.2 Given = − − ,calculate the circulation of F around a
closed path shown.
• = −
EXER
CISE
3.2 Calculate the circulation of= cos ∅ + sin ∅
around the edge L of thewedge defined by 0 ≤ ≤ 2,0 ≤ ∅ ≤ 60°, = 0 and
shown.
• =
DEL OPERATOR
Del Operator
The del operator, written as ,is a vector differential operator.
In Cartesian coordinates,
a.k.a gradient operator
Del Operator
It is not a vector in itself,Useful in defining the following:
1. The gradient of a scalar V, ( )2. The divergence of a vector A,( • )3. The curl of a vector A, ( × )4. The Laplacian of a scalar V, ( )
Del Operator
In Cylindrical coordinates,
In Spherical coordinates,
Gradient of a Scalar
The gradient of ascalar field V is a
vector that representsboth the magnitudeand the direction ofthe maximum spacerate of increase of V.
Gradient of a ScalarIn Cartesiancoordinates,
In Cylindrical coordinates,
Gradient of a ScalarIn Spherical coordinates,
Gradient of a ScalarThe followingcomputationformulas on
gradient,which are
easilyproved,
should benoted:
where U and V arescalars and n is an
integer
EXAM
PLE 3
.3 Find the gradient of the followingscalar fields:a) =b) =c) =
EXER
CISE
3.3 Determine the gradient of the
following scalar fields:a) = +b) = + ∅ +c) = + ∅
EXAM
PLE 3
.4 Given = + ,compute and the direction
derivative / in the direction+4 + 12 at (2, -1, 0).
−
EXER
CISE
3.4 Given = + + . Find
gradient at point (1, 2, 3) and thedirectional derivative of at the same
point in the direction toward point(3, 4, 4).
+ + ,
EXAM
PLE 3
.5 Find the angle at which line= =intersects the ellipsoid+ + 2 = 10.
= .
EXER
CISE
3.5 Calculate the angle between the
normal to the surfaces+ = 3 and log − = −4at the point of intersection (-1, 2, 1)
. °
Divergence of a Vector and Divergence Theorem
The divergence of A at a given point P isthe outward flux per unit volume as the
volume about P.
where ∆ is the volume enclosed by theclosed surface S in which P is located
Divergence of a Vector and Divergence Theorem
a) The divergence of a vector field at point P is positivebecause the vector diverges (spreads out) at P.
b) A vector field has negative divergence (convergence)at P
c) A vector field has zero divergence at P.
Divergence of a Vector and Divergence Theorem
In Cartesian coordinates,
In Cylindrical coordinates,
Divergence of a Vector and Divergence Theorem
In Spherical coordinates,
Note the following properties of thedivergence of a vector field:
Divergence of a Vector and Divergence Theorem
The divergence theorem states that thetotal outward flux of a vector field A
through a closed surface S is the same asthe volume integral of the divergence of
A.
Otherwise known asGauss-Ostrogradsky theorem
EXAM
PLE 3
.6 Find the divergence of these vectorfields:a) = +b) = + ∅ +c) = + +∅
EXER
CISE
3.6 Determine the divergence of the
following vector fields and evaluatethem at the specified points,:
EXAM
PLE 3
.7 If = 10 ( + ),determine the flux of G outof the entire surface of the
cylinder= 1, 0 ≤ ≤ 1.Confirm the result usingthe divergence theorem.
Ψ =
EXER
CISE
3.7 Determine the flux of= cos + sin
over the closed surface of the cylinder0 ≤ ≤ 1, = 4.Verify the divergence theorem for this case.
Curl of a Vector and Stoke’s TheoremThe curl of A is an axial (or rotational)
vector whose magnitude is themaximum circulation of A per unit area
as the area tends to zero and whosedirection is the normal direction of thearea when the area is oriented so as to
make a circulation maximum.
Curl of a Vector and Stoke’s Theorem
In Cartesiancoordinates
Curl of a Vector and Stoke’s Theorem
In Cylindricalcoordinates
Curl of a Vector and Stoke’s TheoremIn Spherical coordinates
Curl of a Vector and Stoke’s Theorem
Note the following properties of the curl:
Curl of a Vector and Stoke’s Theorem
The curl provides the maximum value ofthe circulation of the field per unit area
(or circulation density) and indicates thedirection to which this maximum value
occurs.
Measures the circulation of how muchthe field curls around P
Curl of a Vector and Stoke’s Theorem
a) Curl at P points out of the pageb) Curl at P is zero.
Curl of a Vector and Stoke’s TheoremStokes’s theorem statesthat the circulation of avector field A around a(closed) path L is equalto the surface integral
of the curl of A over theopen surface Sbounded by L
provided that A and× are continuouson S.
EXAM
PLE 3
.8 Determine the curl of these vectorfields:a) = +b) = + ∅ +c) = + +∅
EXER
CISE
3.8 Determine the curl of the following
vector fields and evaluate them at thespecified points,:
EXAM
PLE 3
.9 If = + ∅, evaluate∮ • around the path shown.Confirm using Stokes’s theorem.
4.941
EXER
CISE
3.9 Confirm the circulation of= cos ∅ + sin ∅
around the edge L of the wedgedefined by 0 ≤ ≤ 2, 0 ≤ ∅ ≤ 60°, =0 and shown using Stoke’s Theorem
• =
EXAM
PLE 3
.10 For the vector field A,show explicitly that
× = 0;that is, the divergence of the curl of
any vector field is zero.
EXER
CISE
3.10 For a scalar field V, show that× = 0;
that is, the curl of the gradient ofany scalar field vanishes.
Laplacian of a ScalarThe Laplacian of a scalar field V, writtenas , is the divergence of the gradientof V.In Cartesian:
Laplacian of a Scalar
In Cylindrical coordinates,
In Spherical coordinates,
EXAM
PLE 3
.11Find the Laplacian of the followingscalar fields:a) =b) =c) =
EXER
CISE
3.11 Determine the Laplacian of the
following scalar fields:a) = +b) = + ∅ +c) = + ∅
Classification of Vector FieldsA vector field is uniquely characterized by its
divergence and curl.
Neither the divergence nor curl of a vectorfield is sufficient to completely describe a
field.
All vector fields can be classified in terms oftheir vanishing or non vanishing divergence or
curl
Classification of Vector Fields
Classification of Vector FieldsA vector field A is said to be solenoidal
(or divergenceless) if • =Examples:• incompressible fluids,• magnetic fields• conduction current density
under steady state conditions.
Classification of Vector FieldsA vector field A is said to be irrotational
(or potential) if × =Also known as conservative field.
Examples:• electrostatic field• gravitational field
SUMMARY
Reference:Elements of Electromagnetics
by Matthew N. O. Sadiku