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Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

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Surface Area And Surface Integrals

A smooth surface S is called Orientable or two sided if it is possible to define a field n of unit normal vectors on S that varies continuously with position. Spheres and their smooth closed surface in space are orientable. By convention, we choose n on a closed surface to point outward.

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

The surface together with its normal field is called oriented surface. Unit normal is at any point of it is called the positive direction at the point.
Let f(x, y, z) = c be any oriented surface S and R be its projection (Shadow) on coordinate plane (say xy-plane)
The angle between S and R is given by
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
{z-axis is normal to xy-plane}
area of the surface f(x, y, z) = c over a closed and bounded plane region R is

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - MathematicsdS cos θ = dA
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Surface Integral

If R be the projection (shadow) on xy-plane of surface S defined by equation f(x, y, z) = c and g is a continuous function defined at the points of S, then the integral of g over S is the integral.
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Suppose that F is a continuous vector field defined over an oriented surface S and that Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics is unit normal to the surface S. The surface integral of the normal component of F, i.e. F. Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics, is the flux of F across A in the positive direction and is given by
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics(∇ f.k ≠ 0)

Remark :

(a) To find the surface integral of a vector function or flux means the same thing.
(b) If S is a surface defined by a function z = f(x, y) that has continuous first order partial derivatives throughout a region R in the xy-plane then
Let φ(x, y, z) = f(x, y) - z be the level surface

(a) Unit normal to φ(x, y, z) is
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(b) Area of surface S is
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Evaluate Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where A = 18zi + 12j + 3yk and S is the part of the plane 2x + 3y + 6z = 12 which is located in the first octant.

We know that ds = dA/cosθ
where θ is angle between the normal to surface S and normal to its projection in xy-plane

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - MathematicsVector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
S : f(x, y, z) = 2x + 3y + 6z - 12
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Now Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 432 - 360 + 96
= 72 + 96 = 168

Example : Evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where F = yi + (x - 2xz)j - xyk and S is the surface of the sphere x2 + y2 + z2 = a2 above the xy-plane.

F = yi + (x - 2xz)j - xyk
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

= i (-x + 2x) - j (-y - 0) + k(1 - 2z - 1)
= xi + yj - 2zk ...(i)
Let S be the surface given by f(x, y, z) = x2 + y2 + z2 - a2 above xy-plane.
Thus, the projection on xy-plane is the circle x2 + y2 = a2. Also
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics= unit normal to surface S
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= (1/a)(xi + yj + zk) ...(ii)
and Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(iii)
Now Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= π/2(a3 - a3 + a3 - a3) = 0

Example : Evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics over the entire surface S of the region bounded by the cylinder x2 + z2 = 9, x = 0, y = 0, z = 0 and y = 8 where A = 6zi + (2x + y)j - xk.

Here the entire surface S consists of 5 surfaces, namely. S1 : lateral surface of the cylinder ABCD, S2 : AOED, S3 : OBCE, S4 : OAB, S5 : CDE.
Thus, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= l1 + l2 + l3 + l4 + l5(say)
S1 : ABCD : The curved surface S1 is f = x2 + z2 = 9. The unit outward normal to S1 is
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 1/3(6xz - xz) = (5/3)xz
and Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
=  (5 x 9 x 8)/2 = 180
S2 : AOED : The surface S2 is xy-plane i.e. z = 0. Unit outward normal to the surface is Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
S3 : OBCE : Surface S3 is yz-plane i.e. x = 0. Unit outward normal to S3 is Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
S4 : OAB : The section OAB is in xz-plane i.e. y = 0. The unit outward normal to S4 is Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= (2/3)(-27) = -18
S5 : CDE : The section S. is parallel to xz-plane, y = 8. The unit outward normal to S5 is Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 18(1+π)
Thus, the required surface integral is
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Find the flux of the vector field A = (x - 2z)i + (x + 3y + z)j + (5x + y)k through the upper side of the triangle ABC with vertices at the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1).

Equation of the plane containing the given triangle ABC is f(x, y, z) ≡ x + y + z - 1
Unit normal h to ABC is
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

= (1/√3) (i + j + k)
Flux of A = Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Find the surface area of the plane 2z + x + 2y = 12 cut off by x = 0, y = 0, x = 1, Y = 1.

Let φ(x, y, z) = ((12 - x - 2y)/2)-z = 0 be the level surface.
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= (3/2)(y)10
= 3/2

Volume Integral

Let V be a region is space enclosed by a closed surface S. Let F be a vector point function, and φ be a scalar point function, then the triple integralsVector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics are known as volume integral or space integrals.
In component form Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : If V is the region of first octant bounded by y2 + z2 = 9 and the plane x = 2 and F = 2x2 yi - y2 j + 4xz2k. Then evaluate Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 4xy - 2y + 8xz
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : If F = 2z i + xj + yk. Evaluate Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where V is the region bounded by the surfaces x = 0, y = 0, x = 2, y = 4, z = x2, z = 2.

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 32/15(3i + 5k)

Example : Evaluate Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where A = (x + 2y)i - 3z j + xk and V is the closed region in the first octant bounded by the plane 2x + 2y + z = 4

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 3i - j - 2k

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= (8/3)(3i - j - 2k)

Example : Find the volume enclosed between the two surfaces S1 : z = 8 - x2 - y2 and S2 : z  = x2 + 3Y2.

Eliminating z from the two surfaces S1 and S2, we get
8 - x2 - y2 = x2 + 3y2 
⇒ x2 + 2y2 = 4
Thus, the two surface intersect on the elliptic cylinder.
So the solid region between S1 and Sis covered when z varies from x2 + 3y2 to 8 - x2 - y2.
y varies from Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
x varies from - 2 to 2.
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Gauss's Divergence Theorem

Statement : The normal surface integral of a function F over the boundary of a closed region is equal to the volume integral of div F taking throughout the region.
If F is a continuously differentiable vector point function in a region V enclosed by the closed surface S, then :
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
where Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics is the unit outward drawn normal vector to the surface S.
The Cartesian form of (1) is as follows :
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(2)
where F1, F2 and F3 are scalar components of F along the axes.
Remark : The volume integral is reduced into surface integral with the help of this theorem. Conversely, the surface integral can also be reduced into volume integral.
Proof. Let V be the region wrt the axes such that if any straight line is drawn parallel to any axis, then that will meet the surface S in two points only.
In fig., Ais the projection of region V on the plane XOY. Let the coordinates of any point R in the plane A3 be (x, y, 0).
Let any line through R meets the surfaces at the points P and Q.
Therefore z-co-ordinate of Q by ψ(x, y) and that of P be φ(x, y).
Since RP > RQ, therefore φ(x, y) > ψ(x, y)

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Now Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(3)
Let S1 and S2 be the sub-parts of the surface S.
Let Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics be the outwards drawn unit normal vector at any point on the surface S.
Therefore dx dy = dS cos θ = Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics.k dS,
where θ is the angle of the normal with x-axis.
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(4)
and Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(5)
The negative sign of equation (5) shows that the normal drawn outwards make an obtuse angle with z-axis.
The positive sign in equation (4) is because the outwards normal on the surface S1 make an acute angle with z-axis.
Substituting the value from the equations (4) and (5) in (3)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(6)
Similarly,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(7)
and Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(8)
Adding the equations (6), (7) and (8),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

This theorem is also true for the region V enclosed by two closed curves S1 and S2, where the surface S2 is situated in side the surface S1.
Remark : The volume integral is reduced into surface integral with the help of this theorem. Conversely, the surface integral can also be reduced into volume integral.

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where F = 4xz i - y2j + yz k S is the surface of the cube bounded by the planes :

By Gauss’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics [∵ dv = dx dy dz]
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where S is the part of the sphere x2 + y2 + z2 = 1 above the xy plane. 

F = y2z2i + z2x2j + z2y2k
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 2zy2
By Gauss’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - MathematicsVector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
This is the part of the sphere x2 + y2 + z2 = 1 above the xy plane i.e., hemisphere above the xy plane and the limits for x, y, z are as follows :
First integrate wrt z from z = 0 to z = Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics then integrate wrt y, between the limit Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics and finally wrt x between the limits x = -1 to x = 1.
∴ Required integral = Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
put x = sin θ ⇒ dx = cosθ dθ
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Use Gauss’s divergence theorem to show that : 

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where the surface S is the sphere x2 + y2 + z2 = a2

We know that
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
where F1 = x, F2 = y, and F3 = z
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
∴ From equation (1)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics  (1 + 1 + 1)dx dy dz, enclosed by surface S, where V is volume.
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics dx dy dz = 3 (volume of the sphere)
= 3.(4/3)πa2 = 4πa3

Example : If V is the volume enclosed by any closed surface S; show that:
(a) Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(b) Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(c) Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(d) Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

(a) If F be any constant vector, then
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics  [∵ F is a constant vector]
By Gauss's theorem, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
But Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
or, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics [∵ F is a constant vector]
(b) If F be any constant vector, then
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (by interchanging the scalar and vector products)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics [by Gauss’s theorem] ...(1)
But ∇.(F x r) = r.(∇ x F) - F.(∇ x r) = 0 [∵ V x f = 0, ∵ F is constant and ∇ x r = 0]
By (1), Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
or, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(c) By Gauss’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 0 [∵ ∇.(Z∇ x F) = 0]
(d) By Gauss’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (∵ ∇.r = 3)
= 3V

Example : Evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - MathematicsF = zi + xj + 3y2z k, where S is the surface of the cylinder x2 + y2 = 16 include in the first octant between z = 0 and z = 5.

The projection of the given surface on xz plane will be a rectangle whose sides will be x, z axes and lines parallel to these.
Therefore the given integral
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
Normal vector on the given surface x2 + y2 = 16
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 1/4(xz + xy)
Now by (1),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 5 x 8 - (25/2)(0 - 4) = 90

Example : If V is the volume enclosed by any closed surface S; show that : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where F = xi + 2yj + 3zk 

Given F = xi + 2yj + 3zk
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 1 + 2 + 3 = 6
∴ By Gauss’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Verify Gauss’s Divergence theorem and show that : 

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where F = (x3 - yz)i - 2x2yj + 2k S is the surface of the cube bounded by the co-ordinate planes : x = y = z = 0; x = y = z = a 

Here Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 3x2 - 2x2 + 0 = x2
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Verification by direct Integration : In figure outward unit normal have been shown on each face of the cube. Now on the face ABCD Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics= i, x = a, dS = dy dz
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics(∵ x = a)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= a5 - (1/4)a4
Taking Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = - i on the face OEFC,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
∴ On this face x = 0
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Taking Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = j on the face BEFC.
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics [∵ on this face y = a]
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Taking Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = k on the face CDGF,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Taking Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = -k on the face ABEO,

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= -2x2
Adding the integrals on all the faces,
= a5 - (1/4)a4 + (1/4)a4 - (2/3)a5 + 0 + 2a2 - 2a2
= (1/3)a5

Example : Evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematicswhere F = xi + yj + z2k. where S is the closed surface bounded by the cone x2 + y2 = z2 and the plane z = 1.

Here Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 1 + 1 + 2z = 2(1 + z)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

where V is the volume of the given cone and Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics is the centre of gravity of the cone (situated on the axis of the one) which is at a distance y from the vertex O. The base of the given cone is a circle whose radius is 1 and height is also 1.
∴ OG = 3/4, V = (1 /3)π12 . 1 = π/3
∴ Required integral = 2V(1 + Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics 

Stoke's Theorem

Statement : The line integral of a vector function F around any closed curve is equal to the surface integral of curl F taken over any surface of which of the curve is a boundary edge.
If F be any continuous differentiable vector function and S is the surface enclosed by a curve C, then :
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
where n is the unit normal vector at any point of S and drawn in the sense in which a right handed screw would move rotated in the sense of description of C.

Cartesian form :

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics [∵ dz = 0]
Since Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = k, therefore
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Proof, (a) For surfaces in the planes :

Let the region S1 be subdivided into sub-regions Si such that if a straight line is drawn parallel to any axis then that will meet the curve Ci atmost in two points.
Let Ci be situated between the lines x = a and x = b.
Let any line the be drawn parallel to y-axis meets the curve at the point P and Q, where the ordinate of P is y = φ(x) and that of Q is y = ψ(x). In the figure limit of the curve has been divided by the line PQ in two parts C1 and C2
Now,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics...(1)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Similarly, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(2)
From (1) and (2),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
or, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(b) The Cartesian form of the surface in space,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics [*by Jacobian] ...(1)
where D is any region in u-v plane whose image is S.
Taking the terms of F1 in the equation (1),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (where the additional last term = 0)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Similarly, the values of the terms F2 and F3 in (1) are respectively
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Therefore the RHS of (1)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Now by Stoke’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Similarly,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
and
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Therefore Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Now applying Stoke’s theorem for each subregion and adding the results,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Verify Stoke’s theorem for the function F = zi + xj + yk, where the curve C is the unit circle in the xy-plane bounding the hemisphere z = Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Given F = zi + xj - yk and r = xi + yj + zk
∴ F.dr = (zi + xj + yk).(idx + j dy + k dz)  
or, F.dr = z dx + x dy + y dz ...(1)
The unit circle C in xy plane x2 + y2 = 1, z = 0
∴ For the curve C, z = 0 and dz = 0
∴ by (1), F.dr = x dy ...(2)
From the fig., x = cos φ, y = sin φ, where φ varies from 0 to 2π.
∴ From (2),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 1/2[2π] = π ...(3)
From the fig, the direction cosines of the line OP are sin θ cos φ, sin θ sinφ, cos θ
Therefore Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = (sin θ cos φ)i + (sin θ sin φ)j + (cos θ)k,
where Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics is outer normal at the point P.
Now Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= i + j + k
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics •(∇ x F) = [(sin θ cos φ)i + (sin θ sin φ)j + (cos θ)k]•[i + j + k]
= sin θ cos φ + sin θ sin φ + cos θ
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where dS = sin θ dφ dθ
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(4)
Therefore from (3) and (4),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Using Stoke’s theorem, evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where C is the square in the xy plane with vertices respectively : (1, 0), (-1, 0), (0, 1); (0, 1)

Writing Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where dr = xi + yj
Here F = xy i + xy2 j
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Taking Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = k and dS = dx dy,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics•(∇ x F) = k•(y2 -x)k = y2 - x
By Stokes’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Verify Stoke’s theorem for the function F = x2i + xy j integrated round the square in the plane z = 0, whose sides are along the lines x = y = 0 and x = y = 0.

In the plane z = 0, r = xi + yj
Therefore
dr = i dx + j dy
∴ F•dr = (x3i + xy j)•(i dx + j dy)
= x2 dx + xy dy
Now Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

(a) On the line AB, y = 0, Therefore dy = 0
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics...(1)
(b) On the line BC, x = a, Therefore dx = 0
∴ Along the side BC of the square Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics...(2)
(c) On the line CD, y = a, Therefore dy = 0
∴ Along the side CD of the square Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(3)
(d) On the line DA, x = 0, Therefore dx = 0
∴ Along the side DA of the square Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(4)
Therefore from (1), (2), (3) and (4), along the square ABCD
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(5)
Check : curl F = Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
On square ABCD, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = k, Therefore curl FVector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = yk•k = y
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(6)
From (5) and (6), Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Hence the Stoke’s theorem is verified.

Example : Evaluate by Stoke’s theorem Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where C is the curve x2 + y2 = 4 and z = 2.

Writing Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
or Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where dr = i dx + j dy + k dz
Therefore taking F = exi + 2y j - k and by Stoke’s theorem
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
where the surface S is the boundary C of the curves, x2 + y2 = 4 and z = 2.
Here C is a circle x2 + y2 = 4 and z = 2 whose centre is D(0, 0, 2) and radius is 2 and C is also the boundary of S1, therefore Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics = k
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(2)
Now Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= i(0) + j(0) + k(0) = 0

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - MathematicsFrom (2), Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics•(∇ x F) = 0
Therefore from (1), the given integral
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Prove by Stoke’s theorem :
(a) Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(b) Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

(a) By Stoke’s theorem
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
Let F = ∇(φψ), therefore ∇ x f = ∇ x (∇φψ)
or, ∇ x f = 0 [∵ ∇ x (∇φψ) = 0]
Therefore from (1),Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics...(2)
and Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
From (2), Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
or, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
(b) Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (By Stoke's theorem) ...(1)
Let F = φ∇ψ
∴ ∇ x F = ∇ x (φ∇ψ)
= (∇φ) x ∇ψ + φ∇ x ∇ψ [∵ ∇ x fV = (∇φ) x V + φ(∇ x V)]
or, ∇ x F = ∇φ x ∇ψ [∵ ∇ x (∇ψ) = 0] ...(2)
Therefore from (1) and (2)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Green's Theorem

Statement: If φ and ψ are two continuously differentiable vector point functions such that ∇φ and ∇ψ are also continuously differentiable within V enclosed by a surface S, then
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Proof. Using Gauss theorem for the vector point function φ∇ψ
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
But  Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - MathematicsVector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(2)
From (1) and (2),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics...(3)
Interchanging φ and ψ,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(4)
Subtracting (4) and (3),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(5)
which is the Green’s theorem

Another Form :
Since Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
where ∂ψ/∂η is the derivative of ψ in the direction of outer normal at the point ψ.
Therefore, Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
From (5), Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Cartesian form of Green’s Theorem :

If C is a regular closed curve in xy plane enclosing a region S, P(x, y) and Q(x, y) be two continuously differentiable functions in the region S, then :
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Let F = Pi + Qj, Since n = k, therefore
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
and Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (P dx + Q dy) ...(2)
From (1) and (2),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(3)
Cor. : When Q = x and P = -y, then Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Therefore from (3), Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (If A is the area of the enclosed region by the curve C)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Evaluate by Green’s theorem : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where C is the rectangle with vertices (π, 0), (0, 0), (π, π/2) and (0, π/2)

Here P = e-x sin y; Q = e-x cos y
∴ ∂P/∂Y = e-x cosy and ∂Q/∂x =  -e-xcos y
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics  [- cosy - cosy] = -2e-x cos y
Therefore by Green’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Use Green’s theorem to evaluate : 

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where C is the triangle enclosed by the lines y = 0, x = π/2 and y = 2x/π. 

Given P = y - sin x and Q = cos x
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
By Green’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
Substituting the values of P and Q in (1) and simplifying
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics 
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics where S is the area of the triangle.
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Evaluate by Green’s theorem : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (x2 - cosh y) dx + (y + sin x )dy,
where C is the rectangle with vertices (0, 0), (π, 0), (π, 1) and (0, 1).

Taking the given points as A, B, C and D respectively, then a rectangle ABCD is obtained as given in the fig.
Here P = x2 - cosh y ⇒ Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
and Q = y + sin x Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Now by Green’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= π cosh 1 - π = π[cosh 1 - 1]
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Example : Evaluate : Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics  [(3x2 - 8y2)dx + (4y - 6xy)dy], where C is the region bounded by the parabolas y = √x and y = x2. Also verify Green’s theorem.

Solving the equations of the parabola y =√x and y = xthe coordinates of the points of intersection are O(0, 0) and A(1, 1)
Here P = 3x2 - 8y2 and Q = 4y - 6xy
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
By Green’s theorem,
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
= 3/2 ...(1)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics

Verification : The line integral towards C1 (towards y = x2)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
and line integral towards C2 (towards y2 = x)
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Required integral = Line integral towards C1 and C2
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(2)
Therefore from (1) and (2), the Green’s theorem is verified.

Example : Show that the area bounded by a simple closed curve C is given by Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics Hence find the whole area of the ellipse.

Here P = -y and Q = x
Therefore Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
By Green’s theorem, we know that
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(1)
Substituting the values of Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics and Q in (1),
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
or Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics ...(2)
Therefore the enclosed area by the simple closed curve C = 1/2Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics (x dy-y dx)
The given curve is an ellipse whose parametric equations are
x = a cosθ and y = b sin θ.
Therefore x = a cosθ and y = b sinθ
∴ The required area of the ellipse
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics
Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics 

The document Vector Calculus - (Part - 2) - Notes | Study IIT JAM Mathematics - Mathematics is a part of the Mathematics Course IIT JAM Mathematics.
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