Test: Green's Theorem - Civil Engineering (CE) MCQ

Concept:

Green's Theorem:-  If two functions M(x, y) and N(x, y and their partial derivatives are single valued and continuous over a region R  bounded by a closed curve C, then

Green Theorem is useful for evaluating a line integral around a closed curve C.

Calculation:

We have,

⇒ 

On comparing, we get

⇒ M = cos x sin y - x y

⇒ N = sin x cos y

On differentiating M partially with respect to 'y'

⇒ ∂M/∂y = cosx cosy − x

On differentiating N partially with respect to 'x'

⇒ Let 1 - x² = t

⇒ On differentiating, we get

⇒ - 2x dx = dt

⇒ x dx = −dt/2

When x = -1

⇒ 1 - 1 = 0 = t

When x = 1

⇒ 1 - 1 = 0 = t

⇒ 

⇒ 2 × 0

⇒ 0

Concept:

Green's theorem converts the line integral to a double integral. 

Green's theorem transform the line integral in xy - plane to a surface integral on the same xy - plane

If P and Q are functions of (x, y) defined in an open region then

Application:

Given the differential equation is:

y dy = -2 xdx

C = 1

∴ The equation of a curve is:

This is an ellipse.

On comparing with  we get:

a = 1 and b = √2

Now, the given integral is:

(4y – 3x)dx + (3x + 2y) dy

Here M = 4y – 3x and N = 3x + 2y

Applying Greens theorem, we convert the line integral to a double integral, i.e.

∴ The given integral becomes:

= 

= - (Area under ellipse)

= 

= 

Green's theorem:

It gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. 

Let R be a closed bounded region in the xy plane whose boundary C consists of finitely many smooth curves.

Let F₁(x, y) & F₂(x, y) be functions that are continuous and have continuous partial derivatives:

 then

According to Green’s theorem

Stoke's theorem:

Stokes’s theorem: It states that the circulation of a vector field  around a (closed) path L is

equal to the surface integral of the curl of  over the open surface S bounded by L

 provided and  ∇×  are continuous on S.

Green's theorem is the two-dimensional special case of Stokes' theorem.

Green's theorem

• It converts the line integral to a double integral. 
• It transforms the line integral in xy - plane to a surface integral on the same xy - plane.

If M and N are functions of (x, y) defined in an open region then from Green's theorem

Concept:

If M(x,y), N(x,y), ∂N/∂y and ∂M/∂x be continuous functions over region R bounded by a simple closed curve c in x-y plane, then according to this theorem:

It is used to simplify the vector integration.

It gives the relation between the closed line and open surface integration.

Calculation:

Given:

Comparing with the standard equation Mdx + Ndy; M = -y² and N = xy.

∴ 

= 3/2

Concept:

Green’s theorem states that:

Given, , i.e

P = -y

Q = x

= 

⇒ 

= 2 × Area enclosed

= 2 [Area of rectangle + Area of semicircle]

= 

⇒ 12 + π

Concept:

Green’s theorem:

Let R be a closed bounded region in the xy plane whose boundary C consists of finitely many

smooth curves.

Let F₁(x, y) & F(x, y) be functions that are continuous and have continuous partial

derivatives 
 
∂F₁ / ∂y and ∂F₂ / ∂x. Then

Analysis:

Given curve C: x² + y² = 1

= 0

The Green’s theorem states that if L and M are functions of (x,y) in an open region containing D and having continuous partial derivatives then,
∫ (F dx + G dy) = ∫∫(dG/dx – dF/dy)dx dy, with path taken anticlockwise.

The Green’s theorem calculates the area traversed by the functions in the region in the anticlockwise direction. This converts the line integral to surface integral.

The Green’s theorem is a special case of the Kelvin- Stokes theorem, when applied to a region in the x-y plane. It is a widely used theorem in mathematics and physics.