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VECTOR CALCULUS Prepared by Engr. Mark Angelo C. Purio

Differential Length, Area, and Volume

Differential Length, Area, and Volume Differential elements in length, area, and volume are useful in vector calculus. They are defined in the Cartesian, Cylindrical, and Spherical coordinate systems

Differential Length, Area, and Volume A. Cartesian Coordinates 1. Differential Displacement

2. Differential Normal Area 3. Differential Volume

Differential Length, Area, and Volume

Differential Elements in the Right-handed Cartesian Coordinate System

Differential Length, Area, and Volume

Differential Normal Areas in Cartesian Coordinates

Differential Length, Area, and Volume A. Cylindrical Coordinates 1. Differential Displacement

2. Differential Normal Area 3. Differential Volume

Differential Length, Area, and Volume

Differential Elements in Cylindrical Coordinates

Differential Length, Area, and Volume

Differential Normal Areas in Cylindrical Coordinates

Differential Length, Area, and Volume A. Spherical Coordinates 1. Differential Displacement

2. Differential Normal Area 3. Differential Volume

Differential Length, Area, and Volume

Differential Elements in Spherical Coordinates

Differential Length, Area, and Volume

Differential Normal Areas in Spherical Coordinates

EXAMPLE 3.1

Consider the object shown. Calculate: a) b) c) d) e) f)

The distance BC The distance CD The surface area ABCD The surface area ABO The surface area AOFD The volume ABDCFO a) 10 b) 2.5 π c) 25 π

d) 6.25 π e) 50 f) 62.5 π

EXERCISE 3.1

Disregard the differential lengths and imagine that the object is part of a spherical shell. It may be describe as 3 ≤ ≤ 5, 60° ≤ ≤ 90°, 45° ≤ ∅ ≤ 60° where surface = 3 is the same as , surface = 60° is , and surface ∅ = 45° is . Calculate a) The distance DH b) The distance FG c) The surface area AEHD a) 0.7854 d) The surface area ABDC b) 2.618 e) The volume of the c) 1.179 object

d) 4.189 e) 4.276

Line, Surface, and Volume Integrals

Line, Surface, and Volume Integrals By a line we mean the path along a curve in space. Line, curve, and contour can be used interchangeably. The line integral ∫ • is the of the tangential component of A along curve L.

Line, Surface, and Volume Integrals Given a vector field A and a curve L, we define the integral

as the line integral of A around L.

Path of integration of a vector field

Line, Surface, and Volume Integrals If the path of integration is a closed curve such as abcba Becomes a closed contour integral

which is called circulation of A around L

Line, Surface, and Volume Integrals Given a vector field A, continuous in a region containing the smooth surface S, we define the surface integral or the flux of A through S as

The flux of a vector field A through surface S

Line, Surface, and Volume Integrals where at any point on S , is the unit normal to S. For a closed surface (defining a volume): The flux of a vector field A through surface S

Which is referred to a the net outward flux of A from S.

Line, Surface, and Volume Integrals We define

as the volume integral of the scalar the volume .

over

The physical meaning of the line, surface, or volume integral depends on the nature of the physicsl quantity represented by A or .

EXAMPLE 3.2

Given = − − , calculate the circulation of F around a closed path shown.

•

=−

EXERCISE 3.2

Calculate the circulation of = cos ∅ + sin ∅

around the edge L of the wedge 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. 2. 3. 4.

The gradient of a scalar V, ( ) The divergence of a vector A,( • ) The curl of a vector A, ( × ) The Laplacian of a scalar V, ( )

Del Operator In Cylindrical coordinates,

In Spherical coordinates,

Gradient of a Scalar The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the maximum space rate of increase of V.

Gradient of a Scalar In Cartesiancoordinates,

In Cylindrical coordinates,

Gradient of a Scalar In Spherical coordinates,

Gradient of a Scalar The following computation formulas on gradient, which are easily proved, should be noted:

where U and V are scalars and n is an integer

EXAMPLE 3.3

Find the gradient of the following scalar fields: a) = b) = c) =

EXERCISE 3.3

Determine the gradient of the following scalar fields: a) = + b) = + ∅+ c) = + ∅

EXAMPLE 3.4

Given = + , compute and the direction derivative / in the direction +4 + 12 at (2, -1, 0).

−

EXERCISE 3.4

Given = + + . Find gradient at point (1, 2, 3) and the directional derivative of at the same point in the direction toward point (3, 4, 4).

+

+

,

EXAMPLE 3.5

Find the angle at which line = = intersects the ellipsoid + + 2 = 10.

=

.

EXERCISE 3.5

Calculate the angle between the normal to the surfaces + = 3 and log − = −4 at the point of intersection (-1, 2, 1)

. °

Divergence of a Vector and Divergence Theorem The divergence of A at a given point P is the outward flux per unit volume as the volume about P.

where ∆ is the volume enclosed by the closed 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 positive because 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 the divergence of a vector field:

Divergence of a Vector and Divergence Theorem The divergence theorem states that the total outward flux of a vector field A through a closed surface S is the same as the volume integral of the divergence of A.

Otherwise known as Gauss-Ostrogradsky theorem

EXAMPLE 3.6

Find the divergence of these vector fields: a) = + b) c)

=

=

+

∅

+

∅

+

+

EXERCISE 3.6

Determine the divergence of the following vector fields and evaluate them at the specified points,:

EXAMPLE 3.7

If

= 10

(

+

),

determine the flux of G out of the entire surface of the cylinder = 1, 0 ≤ ≤ 1. Confirm the result using the divergence theorem.

Ψ=

EXERCISE 3.7

Determine the flux of =

cos

+ sin

over the closed surface of the cylinder 0 ≤ ≤ 1, = 4. Verify the divergence theorem for this case.

Curl of a Vector and Stoke’s Theorem The curl of A is an axial (or rotational) vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make a circulation maximum.

Curl of a Vector and Stoke’s Theorem In Cartesian coordinates

Curl of a Vector and Stoke’s Theorem In Cylindrical coordinates

Curl of a Vector and Stoke’s Theorem In 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 of the circulation of the field per unit area (or circulation density) and indicates the direction to which this maximum value occurs. Measures the circulation of how much the field curls around P

Curl of a Vector and Stoke’s Theorem

a) Curl at P points out of the page b) Curl at P is zero.

Curl of a Vector and Stoke’s Theorem Stokes’s theorem states that the circulation of a vector field A around a (closed) path L is equal to the surface integral of the curl of A over the open surface S bounded by L provided that A and × are continuous on S.

EXAMPLE 3.8

Determine the curl of these vector fields: a) = + b)

=

c)

=

+

∅

+

∅

+

+

EXERCISE 3.8

Determine the curl of the following vector fields and evaluate them at the specified points,:

EXAMPLE 3.9

If = + ∅ , evaluate ∮ • around the path shown. Confirm using Stokes’s theorem.

4.941

EXERCISE 3.9

Confirm the circulation of = cos ∅ + sin ∅

around the edge L of the wedge defined by 0 ≤ ≤ 2, 0 ≤ ∅ ≤ 60°, = 0 and shown using Stoke’s Theorem

•

=

EXAMPLE 3.10

For the vector field A, show explicitly that × = 0; that is, the divergence of the curl of any vector field is zero.

EXERCISE 3.10

For a scalar field V, show that × = 0; that is, the curl of the gradient of any scalar field vanishes.

Laplacian of a Scalar The Laplacian of a scalar field V, written as , is the divergence of the gradient of V. In Cartesian:

Laplacian of a Scalar In Cylindrical coordinates,

In Spherical coordinates,

EXAMPLE 3.11

Find the Laplacian of the following scalar fields:

a) =

b) = c)

=

EXERCISE 3.11

Determine the Laplacian of the following scalar fields: a) = +

b) c)

=

=

+

+

∅+ ∅

Classification of Vector Fields A vector field is uniquely characterized by its divergence and curl. Neither the divergence nor curl of a vector field is sufficient to completely describe a field. All vector fields can be classified in terms of their vanishing or non vanishing divergence or curl

Classification of Vector Fields

Classification of Vector Fields A 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 Fields A 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

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Differential Length, Area, and Volume

Differential Length, Area, and Volume Differential elements in length, area, and volume are useful in vector calculus. They are defined in the Cartesian, Cylindrical, and Spherical coordinate systems

Differential Length, Area, and Volume A. Cartesian Coordinates 1. Differential Displacement

2. Differential Normal Area 3. Differential Volume

Differential Length, Area, and Volume

Differential Elements in the Right-handed Cartesian Coordinate System

Differential Length, Area, and Volume

Differential Normal Areas in Cartesian Coordinates

Differential Length, Area, and Volume A. Cylindrical Coordinates 1. Differential Displacement

2. Differential Normal Area 3. Differential Volume

Differential Length, Area, and Volume

Differential Elements in Cylindrical Coordinates

Differential Length, Area, and Volume

Differential Normal Areas in Cylindrical Coordinates

Differential Length, Area, and Volume A. Spherical Coordinates 1. Differential Displacement

2. Differential Normal Area 3. Differential Volume

Differential Length, Area, and Volume

Differential Elements in Spherical Coordinates

Differential Length, Area, and Volume

Differential Normal Areas in Spherical Coordinates

EXAMPLE 3.1

Consider the object shown. Calculate: a) b) c) d) e) f)

The distance BC The distance CD The surface area ABCD The surface area ABO The surface area AOFD The volume ABDCFO a) 10 b) 2.5 π c) 25 π

d) 6.25 π e) 50 f) 62.5 π

EXERCISE 3.1

Disregard the differential lengths and imagine that the object is part of a spherical shell. It may be describe as 3 ≤ ≤ 5, 60° ≤ ≤ 90°, 45° ≤ ∅ ≤ 60° where surface = 3 is the same as , surface = 60° is , and surface ∅ = 45° is . Calculate a) The distance DH b) The distance FG c) The surface area AEHD a) 0.7854 d) The surface area ABDC b) 2.618 e) The volume of the c) 1.179 object

d) 4.189 e) 4.276

Line, Surface, and Volume Integrals

Line, Surface, and Volume Integrals By a line we mean the path along a curve in space. Line, curve, and contour can be used interchangeably. The line integral ∫ • is the of the tangential component of A along curve L.

Line, Surface, and Volume Integrals Given a vector field A and a curve L, we define the integral

as the line integral of A around L.

Path of integration of a vector field

Line, Surface, and Volume Integrals If the path of integration is a closed curve such as abcba Becomes a closed contour integral

which is called circulation of A around L

Line, Surface, and Volume Integrals Given a vector field A, continuous in a region containing the smooth surface S, we define the surface integral or the flux of A through S as

The flux of a vector field A through surface S

Line, Surface, and Volume Integrals where at any point on S , is the unit normal to S. For a closed surface (defining a volume): The flux of a vector field A through surface S

Which is referred to a the net outward flux of A from S.

Line, Surface, and Volume Integrals We define

as the volume integral of the scalar the volume .

over

The physical meaning of the line, surface, or volume integral depends on the nature of the physicsl quantity represented by A or .

EXAMPLE 3.2

Given = − − , calculate the circulation of F around a closed path shown.

•

=−

EXERCISE 3.2

Calculate the circulation of = cos ∅ + sin ∅

around the edge L of the wedge 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. 2. 3. 4.

The gradient of a scalar V, ( ) The divergence of a vector A,( • ) The curl of a vector A, ( × ) The Laplacian of a scalar V, ( )

Del Operator In Cylindrical coordinates,

In Spherical coordinates,

Gradient of a Scalar The gradient of a scalar field V is a vector that represents both the magnitude and the direction of the maximum space rate of increase of V.

Gradient of a Scalar In Cartesiancoordinates,

In Cylindrical coordinates,

Gradient of a Scalar In Spherical coordinates,

Gradient of a Scalar The following computation formulas on gradient, which are easily proved, should be noted:

where U and V are scalars and n is an integer

EXAMPLE 3.3

Find the gradient of the following scalar fields: a) = b) = c) =

EXERCISE 3.3

Determine the gradient of the following scalar fields: a) = + b) = + ∅+ c) = + ∅

EXAMPLE 3.4

Given = + , compute and the direction derivative / in the direction +4 + 12 at (2, -1, 0).

−

EXERCISE 3.4

Given = + + . Find gradient at point (1, 2, 3) and the directional derivative of at the same point in the direction toward point (3, 4, 4).

+

+

,

EXAMPLE 3.5

Find the angle at which line = = intersects the ellipsoid + + 2 = 10.

=

.

EXERCISE 3.5

Calculate the angle between the normal to the surfaces + = 3 and log − = −4 at the point of intersection (-1, 2, 1)

. °

Divergence of a Vector and Divergence Theorem The divergence of A at a given point P is the outward flux per unit volume as the volume about P.

where ∆ is the volume enclosed by the closed 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 positive because 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 the divergence of a vector field:

Divergence of a Vector and Divergence Theorem The divergence theorem states that the total outward flux of a vector field A through a closed surface S is the same as the volume integral of the divergence of A.

Otherwise known as Gauss-Ostrogradsky theorem

EXAMPLE 3.6

Find the divergence of these vector fields: a) = + b) c)

=

=

+

∅

+

∅

+

+

EXERCISE 3.6

Determine the divergence of the following vector fields and evaluate them at the specified points,:

EXAMPLE 3.7

If

= 10

(

+

),

determine the flux of G out of the entire surface of the cylinder = 1, 0 ≤ ≤ 1. Confirm the result using the divergence theorem.

Ψ=

EXERCISE 3.7

Determine the flux of =

cos

+ sin

over the closed surface of the cylinder 0 ≤ ≤ 1, = 4. Verify the divergence theorem for this case.

Curl of a Vector and Stoke’s Theorem The curl of A is an axial (or rotational) vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make a circulation maximum.

Curl of a Vector and Stoke’s Theorem In Cartesian coordinates

Curl of a Vector and Stoke’s Theorem In Cylindrical coordinates

Curl of a Vector and Stoke’s Theorem In 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 of the circulation of the field per unit area (or circulation density) and indicates the direction to which this maximum value occurs. Measures the circulation of how much the field curls around P

Curl of a Vector and Stoke’s Theorem

a) Curl at P points out of the page b) Curl at P is zero.

Curl of a Vector and Stoke’s Theorem Stokes’s theorem states that the circulation of a vector field A around a (closed) path L is equal to the surface integral of the curl of A over the open surface S bounded by L provided that A and × are continuous on S.

EXAMPLE 3.8

Determine the curl of these vector fields: a) = + b)

=

c)

=

+

∅

+

∅

+

+

EXERCISE 3.8

Determine the curl of the following vector fields and evaluate them at the specified points,:

EXAMPLE 3.9

If = + ∅ , evaluate ∮ • around the path shown. Confirm using Stokes’s theorem.

4.941

EXERCISE 3.9

Confirm the circulation of = cos ∅ + sin ∅

around the edge L of the wedge defined by 0 ≤ ≤ 2, 0 ≤ ∅ ≤ 60°, = 0 and shown using Stoke’s Theorem

•

=

EXAMPLE 3.10

For the vector field A, show explicitly that × = 0; that is, the divergence of the curl of any vector field is zero.

EXERCISE 3.10

For a scalar field V, show that × = 0; that is, the curl of the gradient of any scalar field vanishes.

Laplacian of a Scalar The Laplacian of a scalar field V, written as , is the divergence of the gradient of V. In Cartesian:

Laplacian of a Scalar In Cylindrical coordinates,

In Spherical coordinates,

EXAMPLE 3.11

Find the Laplacian of the following scalar fields:

a) =

b) = c)

=

EXERCISE 3.11

Determine the Laplacian of the following scalar fields: a) = +

b) c)

=

=

+

+

∅+ ∅

Classification of Vector Fields A vector field is uniquely characterized by its divergence and curl. Neither the divergence nor curl of a vector field is sufficient to completely describe a field. All vector fields can be classified in terms of their vanishing or non vanishing divergence or curl

Classification of Vector Fields

Classification of Vector Fields A 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 Fields A 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

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