Unit – 4
Vector Differential Calculus
The derivative of R(t) with respect to t is given by
if the limit exists. If
R(t) = x(t) i + y(t) j + z(t) k
Then
So
Consequently,
Thus
We define a new operator ∇ by
∇=i
+ j
+ k
Example 1:
Find the gradient of the following
= 
y=
= 
.
= 2x+
Example 2:
Consider the surface z= 10-
at (1,-1,7), find a 3d tangent vector that points in the direction of steepest ascent.
Solution:
Let our tangent vector be v= ai+bj+ck.
To find v , we have that,
The direction of steepest ascent of z(x,y) is given by the two-dimensional vector 
.
First we find the x,y components of v,then we find z component of v
Finding x,y components of v,
As the gradient provides the direction of steepest ascent, we compute it:
Thus, a=-2 and b=4, and are seeking a tangent vector
V=-2i+4j+ck.
Example 1:
Compute 
where F= (3x+
and s is the surface of the box such that 0
use outward normal n
Solution:
Writing the given vector fields in a suitable manner for finding divergence
div F =3+2y+x
we use the divergence theorem to convert the surface integral into a triple integral
Where B is the box 0
, 0
We compute the triple integral of div F=3+2y+x over the box B
=
=
= 36+3=39
Example 2:
For F= (
use divergence theorem to evaluate 
where s is the dphere of radius 3 centred at origin.
Solution:
Since div F= 
, the surface integral is equal to the triple integral.
To evaluate the triple integral we can change value of variables to spherical co-ordinates,
The integral is 
=
.For spherical co-ordinates, we know that the jacobian determinant is dV = 
.therefore, the integral is
= 
= 
=
Example: 3
Find the curl of F(x,y,z) = 3
i+2zj-xk
Curl F = 
=
= 
i - 
= (0-2)i-(-1-0)j+(0-0)k
= -2i+j
Example:4
What is the curl of the vector field F= ( x +y +z ,x-y-z,
)?
Solution:
Curl F = 
= 
=
= (2y+1)i-(2x-1)j+(1-1)k
= (2y+1)i+(1-2x)j+0k
= (2y+1, 1-2x,0)
Example:5
Find the curl of F = (
)i +4zj +
Solution:
Curl F=
= 
=(0-4)i-(2x-0)j+(0+1)k
=(-4)i – (2x)j+1k
=(-4,-2x,1)
Example 6:
Determine 
in the direction of 
Solution:
First we calculate gradient of the points,
Here we require a unit vector but by our knowledge it is clear that the given vector is not so we change it into unit vector,
Solenoidal vector formula:
= 0
Ir-rotattional vector formula:
=0
Example: 1
Show that 
= yz
+zx
+xy
Solution:
= yz
+zx
+xy
To prove : 
=0
= 
= 0
Example:2
Prove 
=(
)
+(2ysinx-4)
3x
is irrotational and find its scalar potential.
Solution:
=(
)
+(2ysinx-4)
3x
=
= 
Hence, 
is irrotational
Equating the co-efficient of 
we get,
Example1:
Determine the vector field 
(x,y) = 
is conservative or not
Solution:
Note that the domain 
is the part of R2 in which y>0.Thus, the domain of 
is part of a plane above the x-axis and this domain is simply connected (there are no holes in this region and this region is connected).Therefore, we can use the cross-partial property of conservative fields to
Then 
and thus 
is conservative.
Example 2:
Test for conservative
Solution:
is conservative if 
Where 
f(x,y,z) = x+g(y,z)
∂f/∂z=∂h/∂z=z^2→h(z)=z^3/3+c
→ f(x,y,z) = x + y^2/2+z^3/3+c
∴ F ⃑=∇f hence F ⃑ is conservative.
Example 1:
F(x,y) = 2i + 3j is a conservative vector field.Find a potential function for it.
Solution:
Note : As F(x,y) is conservative,it has a potential function.That is,there is some function ∅(x,y) such that F(x,y)= ∇∅(x,y).
The function ∅(x,y) can be found by integrating each component of
F(x,y)
To find potential function we consider,
2i +3j = 
Thus ,
By integrating we get,
By combining the above equation we get a possible potential function for
Example 2:
If F (x,y) = 
is conservative? If so ,find a 
such that
F(x,y) = 
Solution:
The vector field F(x,y) = u(x,y)i+v(x,y)j is conservative if and only if
We identify(x,y) = 
,from which we can compute the partial derivatives
They are equal so they are conservative.
Finding the potential 
As we proved for conservative now we find the potential function.
Therefore the function 
can be determined by integrating the equations from each component of
=
And combining the results into a single function 
we get,
....(1)
.....(2)
Integrating (1) w.r.to x and (2) w.r.to y
, for any g(y) and h(x)
These both relations are satisfied by
Some vector identities are as follows:
Example 1:
Show that d
Solution:
Let 
, then 
If 
is any scalar point function, then
d
= 
= 
.
= 
Example 2:
If r=
where 
Then show that ,
(1)
(2)
(3)
Solution:
Given 
and r=
i.e. 
+
differentiate w.r.to x partially we get,
(1)
(2)
=f’(r)
= f’(r).
(3)
)
