VECTOR CALCULUS AND LINEAR ALGEBRAPresented by-

Hetul Patel- 1404101160

Jaina Patel - 1404101160

Kinjal Patel - 1404101160

Kunj Patel -1404101160

Matangi Patel -1404101160

CONTENTS

PHYSICAL INTERPRETATION OF GRADIENT CURL DIVERGENCE SOLENOIDAL AND IRROTATIONAL FIELDS DIRECTIONAL DERIVATIVE

GRADIENT OF A SCALAR FIELD

The gradient of a scalar function f(x1 x2 x3 xn) is denoted by nablaf or where nabla (the nabla symbol) denotes the vector differential operator del The notation grad(f) is also commonly used for the gradient

The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v That is

In 3-dimensional cartesian coordinate system it is denoted by

f f ff

x y z

f f f

x y z

i j k

i j k

PHYSICAL INTERPRETATION OF GRADIENT

One is given in terms of the graph of some function z = f(x y) where f is a reasonable function ndash say with continuous first partial derivatives In this case we can think of the graph as a surface whose points have variable heights over the x y ndash plane

An illustration is given below If say we place a marble at some point (x y) on this graph with zero initial force

its motion will trace out a path on the surface and in fact it will choose the direction of steepest descent

This direction of steepest descent is given by the negative of the gradient of f One takes the negative direction because the height is decreasing rather than increasing

PHYSICAL INTERPRETATION OF GRADIENT

Using the language of vector fields we may restate this as follows For the given function f(x y) gravitational force defines a vector field F over the corresponding surface z = f(x y) and the initial velocity of an object at a point (x y) is given mathematically by ndash nablaf(x y)

The gradient also describes directions of maximum change in other contexts For example if we think of f as describing the temperature at a point (x y) then the gradient gives the direction in which the temperature is increasing most rapidly

CURL

In vector calculus the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field

At every point in that field the curl of that point is represented by a vector

The attributes of this vector (length and direction) characterize the rotation at that point

The direction of the curl is the axis of rotation as determined by the right hand rule and the magnitude of the curl is the magnitude of that rotation

CURL

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

CONTENTS

PHYSICAL INTERPRETATION OF GRADIENT CURL DIVERGENCE SOLENOIDAL AND IRROTATIONAL FIELDS DIRECTIONAL DERIVATIVE

GRADIENT OF A SCALAR FIELD

The gradient of a scalar function f(x1 x2 x3 xn) is denoted by nablaf or where nabla (the nabla symbol) denotes the vector differential operator del The notation grad(f) is also commonly used for the gradient

The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v That is

In 3-dimensional cartesian coordinate system it is denoted by

f f ff

x y z

f f f

x y z

i j k

i j k

PHYSICAL INTERPRETATION OF GRADIENT

One is given in terms of the graph of some function z = f(x y) where f is a reasonable function ndash say with continuous first partial derivatives In this case we can think of the graph as a surface whose points have variable heights over the x y ndash plane

An illustration is given below If say we place a marble at some point (x y) on this graph with zero initial force

its motion will trace out a path on the surface and in fact it will choose the direction of steepest descent

This direction of steepest descent is given by the negative of the gradient of f One takes the negative direction because the height is decreasing rather than increasing

PHYSICAL INTERPRETATION OF GRADIENT

Using the language of vector fields we may restate this as follows For the given function f(x y) gravitational force defines a vector field F over the corresponding surface z = f(x y) and the initial velocity of an object at a point (x y) is given mathematically by ndash nablaf(x y)

The gradient also describes directions of maximum change in other contexts For example if we think of f as describing the temperature at a point (x y) then the gradient gives the direction in which the temperature is increasing most rapidly

CURL

In vector calculus the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field

At every point in that field the curl of that point is represented by a vector

The attributes of this vector (length and direction) characterize the rotation at that point

The direction of the curl is the axis of rotation as determined by the right hand rule and the magnitude of the curl is the magnitude of that rotation

CURL

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

GRADIENT OF A SCALAR FIELD

The gradient of a scalar function f(x1 x2 x3 xn) is denoted by nablaf or where nabla (the nabla symbol) denotes the vector differential operator del The notation grad(f) is also commonly used for the gradient

The gradient of f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v That is

In 3-dimensional cartesian coordinate system it is denoted by

f f ff

x y z

f f f

x y z

i j k

i j k

PHYSICAL INTERPRETATION OF GRADIENT

One is given in terms of the graph of some function z = f(x y) where f is a reasonable function ndash say with continuous first partial derivatives In this case we can think of the graph as a surface whose points have variable heights over the x y ndash plane

An illustration is given below If say we place a marble at some point (x y) on this graph with zero initial force

its motion will trace out a path on the surface and in fact it will choose the direction of steepest descent

This direction of steepest descent is given by the negative of the gradient of f One takes the negative direction because the height is decreasing rather than increasing

PHYSICAL INTERPRETATION OF GRADIENT

Using the language of vector fields we may restate this as follows For the given function f(x y) gravitational force defines a vector field F over the corresponding surface z = f(x y) and the initial velocity of an object at a point (x y) is given mathematically by ndash nablaf(x y)

The gradient also describes directions of maximum change in other contexts For example if we think of f as describing the temperature at a point (x y) then the gradient gives the direction in which the temperature is increasing most rapidly

CURL

In vector calculus the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field

At every point in that field the curl of that point is represented by a vector

The attributes of this vector (length and direction) characterize the rotation at that point

The direction of the curl is the axis of rotation as determined by the right hand rule and the magnitude of the curl is the magnitude of that rotation

CURL

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

PHYSICAL INTERPRETATION OF GRADIENT

One is given in terms of the graph of some function z = f(x y) where f is a reasonable function ndash say with continuous first partial derivatives In this case we can think of the graph as a surface whose points have variable heights over the x y ndash plane

An illustration is given below If say we place a marble at some point (x y) on this graph with zero initial force

its motion will trace out a path on the surface and in fact it will choose the direction of steepest descent

This direction of steepest descent is given by the negative of the gradient of f One takes the negative direction because the height is decreasing rather than increasing

PHYSICAL INTERPRETATION OF GRADIENT

Using the language of vector fields we may restate this as follows For the given function f(x y) gravitational force defines a vector field F over the corresponding surface z = f(x y) and the initial velocity of an object at a point (x y) is given mathematically by ndash nablaf(x y)

The gradient also describes directions of maximum change in other contexts For example if we think of f as describing the temperature at a point (x y) then the gradient gives the direction in which the temperature is increasing most rapidly

CURL

In vector calculus the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field

At every point in that field the curl of that point is represented by a vector

The attributes of this vector (length and direction) characterize the rotation at that point

The direction of the curl is the axis of rotation as determined by the right hand rule and the magnitude of the curl is the magnitude of that rotation

CURL

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

PHYSICAL INTERPRETATION OF GRADIENT

Using the language of vector fields we may restate this as follows For the given function f(x y) gravitational force defines a vector field F over the corresponding surface z = f(x y) and the initial velocity of an object at a point (x y) is given mathematically by ndash nablaf(x y)

The gradient also describes directions of maximum change in other contexts For example if we think of f as describing the temperature at a point (x y) then the gradient gives the direction in which the temperature is increasing most rapidly

CURL

In vector calculus the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field

At every point in that field the curl of that point is represented by a vector

The attributes of this vector (length and direction) characterize the rotation at that point

The direction of the curl is the axis of rotation as determined by the right hand rule and the magnitude of the curl is the magnitude of that rotation

CURL

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

CURL

In vector calculus the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field

At every point in that field the curl of that point is represented by a vector

The attributes of this vector (length and direction) characterize the rotation at that point

The direction of the curl is the axis of rotation as determined by the right hand rule and the magnitude of the curl is the magnitude of that rotation

CURL

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

CURL

Definition

It is also defined as

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

POINTS TO BE NOTED

If curl F=0 then F is called an irrotational vector

If F is irrotational then there exists a scalar point function ɸ such that F=nablaɸ where ɸ is called the scalar potential of F

The work done in moving an object from point P to Q in an irrotational field is

= ɸ(Q)- ɸ(P) The curl signifies the angular velocity or

rotation of the body

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

DIVERGENCE

In vector calculus divergence is a vector operator that measures the magnitude of a vector fields source or sink at a given point in terms of a signed scalar

More technically the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

DIVERGENCE

If F = Pi + Q j + R k is a vector field on and partPpartx partQparty and partRpartz exist the divergence of F is the function of three variables defined by

div P Q R

x y z

F

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

DIVERGENCE

In terms of the gradient operator

The divergence of F can be written symbolically as the dot product of and F

x y z

i j k

div F F

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

SOLENOIDAL AND IRROTATIONAL FIELDS

The with null divergence is called solenoidal and the field with null-curl is called irrotational field

The divergence of the curl of any vector field A must be zero ie

nabla (nablatimesA)=0 Which shows that a solenoidal field can be

expressed in terms of the curl of another vector field or that a curly field must be a solenoidal field

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

SOLENOIDAL AND IRROTATIONAL FIELDS

The curl of the gradient of any scalar field ɸ must be zero ie

nabla (nablaɸ)=0 Which shows that an irrotational field can be

expressed in terms of the gradient of another scalar field or a gradient field must be an irrotational field

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

DIRECTIONAL DERIVATIVE

The directional derivative is the rate at which the function changes at a point in the direction It is a vector form of the usual derivative and can be defined as-

= =

Where is called nabla or del and denotes a unit vector

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15

THANK YOU

• Vector calculus and linear algebra
• Contents
• GRADIENT OF A SCALAR FIELD
• PHYSICAL INTERPRETATION OF GRADIENT
• PHYSICAL INTERPRETATION OF GRADIENT (2)
• curl
• Curl
• Points to be noted
• Slide 9
• Slide 10
• Slide 11
• Slide 12
• SOLENOIDAL AND IRROTATIONAL FIELDS
• Directional derivative
• Slide 15