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VECTOR CALCULUS

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10MAT11 UNIT -4
VECTOR CALCULUS
Introduction: Vector is a quantity having both magnitude and direction.
Eg: Force, Velocity.
Scalar and Vector Function:
Scalar point Function: If to every point (x,y,z) of a region R in space there corresponds
a scalar

then is called a scalar point function and we say that a scalar field
is defined in R.
Eg: 1.
 
 
2.
Vector point Function: If to every point (x,y,z) of a region R in space there
corresponds a vector
 then
is called a vector point function and we say
that a vector field
is defined in R.
Eg:1.
 
 
2.
  
Vector function of single variable:
Let the position vector of a point P(x,y,z) in space be 
If x,y,z are all functions of a single parameter t, then is said to be a vector function
of t and also known as Vector point Function and denoted by
Hence
  is called vector equation of the curve.
Now,





 


is a vector along the tangent to the curve at P.
If t is the time variable then


is the velocity of the particle at time t.
Also,

is the rate of change of velocity i.e., it gives acceleration.
Some observations:
Operators:
i) The vector differential operator  read as Nabla or Del is defined by





ii) The Laplacian operator
is defined by



=

Gradient, Divergence,Curl and Laplacian
Gradient of a scalar field: If

is continuously differentiable scalar function
then the gradient of or grad is defined by 
i.e.,grad =  =






Note: grad is a scalar quantity.
Geometrical meaning of the gradient:
If

is a scalar function then grad is a vector normal to the surface

.
Angle between two surfaces:

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Angle between two surfaces is defined to be equal to the angle between their normals.
If

and

be the equations of the two surfaces then





where is the angle between the normals.
Directional Derivative:
If

is a scalar function and
is a given direction then    where

is
called as the directional derivative of along .
Theorem: The directional derivative of a scalar function at any point is maximum
along  and its maximum value is

.
Divergence of a vector field:
If
 is a continuously differentiable vector function then divergence of
or
div
is defined to be 
.
i.e.,If
 
 
where
are all functions of x,y,z then div
= 
=



 
 
=




Note: div
is a scalar quantity.
Physical meaning of divergence: If
 represents any physical quantity, the
divergence of
gives the rate at which the physical quantity is originating at that point
per unit volume.
Curl of a vector field:
If
 is a continuously differentiable vector function then curl of
or curl
is
defined to be   
.
i.e., If
 
 
where
are all functions of x,y,z then curl
=
  
=



=







Note: Curl is a vector quantity.
Physical meaning of curl: The curl f any vector point function will give the measure of
the angular velocity at any point.
Laplacian : If

continuously differentiable scalar function and
 is a
continuously differentiable vector function then
Laplacian of =
=



Laplacian of
=
=



Note: If is a scalar function, the equation
= 0 is called Laplace’s equation and a
function which satisfies Laplace’s equation is called harmonic function.

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10MAT11 UNIT -4 VECTOR CALCULUS Introduction: Vector is a quantity having both magnitude and direction. Eg: Force, Velocity. Scalar and Vector Function: Scalar point Function: If to every point (x,y,z) of a region R in space there corresponds a scalar then is called a scalar point function and we say that a scalar field is defined in R. Eg: 1. 2. Vector point Function: If to every point (x,y,z) of a region R in space there corresponds a vector then is called a vector point function and we say that a vector field is defined in R. Eg:1. 2. Vector function of single variable: Let the position vector of a point P(x,y,z) in space be If x,y,z are all functions of a single parameter t, then is said to be a vector function of t and also known as Vector point Function and denoted by Hence is called vector equation of the curve. Now, is a vector along the tangent to the curve at P. If t is the time variable then is the velocity of the particle at time t. Also, is the rate of change of velocity i.e., it gives acceleration. Some observations: Operators: i) The vector differential operator read as Nabla or Del is defined by ii) The Laplacian operator is defined by = Gradient, Divergence,Curl and Laplacian Gradient of a scalar field: If is continuously differentiable scalar function then the gradient of or grad is defined by i.e.,grad = = Note: grad is a scalar quantity. Geometrical meaning of the gradient: If is a scalar function then gr ...
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