The document summarizes key concepts in vector calculus and linear algebra including: - The gradient of a scalar field describes the direction of steepest ascent/descent and is defined as the vector of partial derivatives. - Curl describes infinitesimal rotation of a 3D vector field and is defined as the cross product of the del operator and the vector field. - Divergence measures the magnitude of a vector field's source or sink and is defined as the del operator dotted with the vector field. - Solenoidal fields have zero divergence and irrotational fields have zero curl. The curl of a gradient is always zero and the divergence of a curl is always zero.

1. VECTOR CALCULUS AND
LINEAR ALGEBRA
Presented by:
Upendra Talar-140410117055
Jahnvi Trivedi-140410117056
Keyur Trivedi-140410117057
Maharsh Trivedi-140410117058
Divya Upadhyay-140410117059
Guided by: Jaimin Patel sir

2. CONTENTS:
oPHYSICAL INTERPRETATION OF
GRADIENT
oCURL
oDIVERGENCE
oSOLENOIDAL AND IRROTATIONAL
FIELDS
oDIRECTIONAL DERIVATIVE

3. GRADIENT OF A SCALAR FIELD:
 The gradient of a scalar function f(x1, x2, x3, ..., xn) is
denoted by ∇f or where ∇ (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 f
f
x y z
f f f
x y z
  
   
  
  
  
  
i j k
i j k

4. 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 – 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 – 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.

5.  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 – ∇f(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.

6. 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.

7.  Definition:
 It is also defined as:

8. 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=∇ɸ 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.

9.  Conservative field: If F is a vector force field then line
integral
Represents the work done around a closed path. If it is
zero then the field is said to be conservative.
 Irrotational field is always conservative. If F is irrotational
then the scalar potential of F is obtained by the formula:

10. DIVERGENCE
 In vector calculus, divergence is a vector operator
that measures the magnitude of a vector
field's 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.

11. DIVERGENCE
 If F = Pi + Q j + R k is a vector field on
and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist,
the divergence of F is the function of three variables
defined by:
Equation 9
div
P Q R
x y z
  
  
  
F
° 3

12. 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
Equation 10

13. DIVERGENCE
If F(x, y, z) = xz i + xyz j – y2 k
find div F.
 By the definition of divergence (Equation 9 or 10) we have:
     2
div
xz xyz y
x y z
z xz
 
  
   
  
 
F F

14. 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, i.e.
∇· (∇×A)=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.

15.  The curl of the gradient of any scalar field ɸ must
be zero , i.e. ,
 ∇ (∇ɸ)=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.

17. THEOREM

19. FIND THE DIRECTIONAL DERIVATIVE OF F AT THE
GIVEN POINT IN THE DIRECTION INDICATED BY THE
ANGLE
4
),1,2(,),( 432 
  yyxyxf