We will cover following topics

Green’s Theorem

Green’s theorem gives a relation between a double integral over a region $R$ in the $x y$ plane and the line integral over a closed curve $C$ enclosing the region $R$ . It helps to evaluate line integral easily.

Statement of Green’s theorem: If $P (x, y)$ and $Q (x, y)$ are continuous functions with continuous partial derivatives in a region $R$ in the $x y$ plane and on its boundary $C$ which is a simple closed curve then $\oint_{C} (P d x + Q d y) = \iint_{R} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d x d y$ where $C$ is described in the anticlockwise sense (which is the positive sense).

The vector form of Green’s theorem is given by $\oint_{C} \vec{F} \cdot d \vec{r} = \iint_{R} \nabla \times \vec{F} \cdot \vec{k} d R$ , where $d R = d x d y$ .

Gauss’s Divergence Theorem

Gauss’s Divergence Theorem is used to convert a volume integral to a surface integral and vice-versa.

Let $V$ be a region in space with piecewise smooth boundary $S$ and let $F$ is a continuously differentiable vector field defined on a neighbourhood of $V$ , then, according to the divergence theorem:

∭_{V} (\nabla \cdot F) d V = \iint_{S} (n \times F) d S

Stokes’ Theorem

Stokes’ Theorem is used to convert a line integral to a surface integral or vice-versa.

According to Stokes’ theorem, if $F$ be a vector-field, then:

\iint_{S} (curl F \cdot \hat{n}) d S = \int_{C} (F \cdot d r)

where $C$ is the boundary of the surface $S$ , and $C$ and $S$ have same orientations.

Green’s Identities

Green’s Identities are a set of three vector identities.

The first of these can be derived from the vector derivative identitities and Gauss’ divergence theorem: The two vector derivative equations are given by:

\nabla \cdot (ψ \nabla ϕ) = ψ \nabla^{2} ϕ + (\nabla ψ) \cdot (\nabla ϕ) (1)

\nabla \cdot (ϕ \nabla ψ) = ϕ \nabla^{2} ψ + (\nabla ϕ) \cdot (\nabla ψ) (2)

From the Gauss’s divergence theorem,

\int_{V} (\nabla \cdot F) d V = \int_{S} F \cdot d a (3)

From (2) and (3), we get:

\int_{S} ϕ (\nabla ψ) \cdot d a = \int_{V} [ϕ \nabla^{2} ψ + (\nabla ϕ) \cdot (\nabla ψ)] d V

Subtracting (2) from (1), we get Green’s First Identity:

\nabla \cdot (ϕ \nabla ψ - ψ \nabla ϕ) = ϕ \nabla^{2} ψ - ψ \nabla^{2} ϕ (4)

From (3) and (4), we get Green’s Second Identity:

\int_{V} (ϕ \nabla^{2} ψ - ψ \nabla^{2} ϕ) d V = \int_{5} (ϕ \nabla ψ - ψ \nabla ϕ) \cdot d a

Let $u$ have continuous first partial derivatives and be harmonic inside the region of integration.

Then, according to Green’s Third Identity:

u (x, y) = \frac{1}{2 π} \oint_{C} [\ln (\frac{1}{r}) \frac{\partial u}{\partial n} - u \frac{\partial}{\partial n} \ln (\frac{1}{r})] d s

PYQs

Gauss’s Divergence Theorem

1) Let $\vec{F} = x y^{2} \hat{i} + (y + x) \hat{j}$ . Integrate $(\nabla \times \vec{F}) \cdot \vec{k}$ over the region in the first quadrant bounded by the curve $y = x^{2}$ and $y = x$ using Green’s theorem.

[2018, 12M]

2) Using Green theorem, evaluate the $\int_{C} F (\vec{r}) . d \vec{r}$ counterclockwise where $F (r) = (x^{2} + y^{2}) \hat{i} + (x^{2} - y^{2}) \hat{j}$ and $d \vec{r} = x \hat{i} + d y \hat{j}$ and the curve $C$ is the boundary of the region $R = {(x, y) | 1 \leq y \leq 2 - x^{2}}$ .

[2017, 8M]

3) Evaluate $\int_{C} e^{- x} (\sin y d x + \cos y d y)$ , where $C$ is the rectangle with vertices $(0, 0)$ , $(π, 0)$ , $(π, \frac{π}{2})$ , $(0, \frac{π}{2})$ .

[2015, 12M]

4) Verify Green’s theorem in the plane for $\oint_{C} [(x y + y^{2}) d x + x^{2} d y]$ where $C$ is the closed curve of the region bounded by $y = x$ and $y = x^{2}$ .

[2012, 20M]

5) Verify Green’s theorem for $e^{- x} \sin y d x + e^{- x} \cos y$ by the path of integration being the boundary of the square whose vertices are $(0, 0)$ , $(\frac{π}{2}, 0)$ , $(\frac{π}{2}, \frac{π}{2})$ and $(0, \frac{π}{2})$ .

[2010, 20M]

Gauss’s Divergence Theorem

1) State Gauss divergence theorem. Verify this theorem for $\vec{F} = 4 x \hat{i} - 2 y^{2} \hat{j} + z^{2} \hat{k}$ , taken over the region bounded by $x^{2} + y^{2} = 4$ , $z = 0$ and $z = 3$ .

[2019, 15M]

2) If $S$ is the surface of the sphere $x^{2} + y^{2} + z^{2} = a^{2}$ , then evaluate

\iint_{S} [(x + y) d y d z + (y + z) d z d x + (x + y) d x d y]

using Gauss’s divergence theorem.

[2018, 12M]

3) Evaluate the integral $\iint_{S} F n d s$ , where $\bar{F} = 3 x y^{2} \hat{i} + (y x^{2} - y^{3}) j + 3 z x^{2} K$ and $S$ is a surface of the cylinder $y^{2} + z^{2} \leq 4, - 3 \leq x \leq 3$ using divergence theorem.

[2017, 9M]

4) By using Divergence Theorem of Gauss, evaluate the surface integral $\iint {(a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2})}^{- \frac{1}{2}} d S$ , where $S$ is the surface $e$ of the ellipsoid $a x^{2} + b y^{2} + c z^{2} = 1$ , $a$ , $b$ and $c$ being all positive constants.

[2013, 15M]

5) Verify Gauss’ Divergence Theorem for the vector $\vec{v} = x^{2} \hat{i} + y^{2} \hat{j} + z^{2} \hat{k}$ taken over the cube $0 \leq x, y, z \leq 1$ .

[2011, 15M]

6) Use the divergence theorem to evaluate $\iint_{s} \vec{V} n d A$ , where $\vec{V} = x^{2} z \vec{i} y \vec{j} - x z^{2} \vec{k}$ and $S$ is he boundary of the region bounded by the paraboloid $z = x^{2} + y^{2}$ and the plane $z = 4 y$ .

[2010, 20M]

7) Using divergence theorem, evaluate $\iint_{s} \bar{A} \cdot d \bar{S}$ where $\bar{A} = x^{3} \hat{i} + y^{3} \hat{j} + z^{3} \hat{k}$ and $S$ is the surface of the sphere $x^{2} + y^{2} + z^{2} = a^{2}$ .

[2009, 20M]

8) Evaluate

\iint_{S} (x^{3} d y d z + x^{2} y d z d x + x^{2} z d x d y)

by Gauss divergence theorem, where $S$ is the surface of the cylinder $x^{2} + y^{2} = a^{2}$ bounded by $z = 0$ and $z = b$ .

[2005, 15M]

9) Verify Gauss’ divergence theorem of $A = (4 x, - 2 y^{2}, z^{2})$ taken over the region bounded by $x^{2} + y^{2} = 4, z = 0$ and $z = 3$ .

[2001, 15M]

Stokes’ Theorem

1) Evalute by Stokes’ theorem $\oint_{C} e^{x} d x + 2 y d y - d z$ , where $C$ is the curve $x^{2} + y^{2} = 4$ , $z = 2$ .

[2019, 5M]

2) Evaluate the line integral $\int_{C} - y^{3} d x + x^{3} d y + z^{3} d z$ using Stokes’ theorem. Here $C$ is the intersection of the cylinder $x^{2} + y^{2} = 1$ and the plane $x + y + z = 1$ . The orientation on $C$ corresponds to counterclockwise motion in the $x y - p l a n e$ .

[2018, 13M]

3) Evaluate by Stokes’ theorem:

\int (y d x + z d y + x d z)

where is the curve given by $x^{2} + y^{2} + z^{2} - 2 a x - 2 a y = 0$ , $x + 2 y = 2 a$ , starting from $(2 a, 0, 0)$ and then going below the z-plane.

[2014, 20M]

4) Use Stokes’ theorem to evaluate the line integral $\int_{C} (- y^{3} d x + x^{3} d y - z^{3} d z)$ , where $C$ is the intersection of the cylinder $x^{2} + y^{2} = 1$ and the plane $x + y + z = 1$ .

[2013, 15M]

5) If $\vec{F} = y \vec{i} + (x - 2 x z) \vec{j} - x y \vec{k}$ , evaluate $\iint_{S} (\vec{\nabla} \times \vec{F}) . \vec{n} d \vec{s}$ , where $S$ is the surface of the sphere $x^{2} + y^{2} + z^{2} = a^{2}$ above the $x y - p l a n e$ .

[2012, 20M]

6) If $\vec{u} = 4 y \hat{i} + x \hat{j} + 2 z \hat{k}$ , calculate the double integral $\iint (\nabla \times \vec{u}) d \vec{s}$ over the hemisphere given by $x^{2} + y^{2} + z^{2} = a^{2}$ , $z \geq 0$ .

[2011, 15M]

7) Find the work done in moving the particle once round the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ , $z = 0$ under the field of force of given by $\bar{F} = (2 x - y + z) \hat{i} + (x + y - z^{2}) \hat{j} + (3 x - 2 y + 4 z) \hat{k}$ .

[2009, 20M]

8) Find the value of $\iint_{s} (\vec{\nabla} \times \vec{f}) . d s$ taken over the upper portion of the surface $x^{2} + y^{2} - 2 a x + a z = 0$ and the bounding curve lies in the plane $z = 0$ , when $\vec{F} = (y^{2} + z^{2} - x^{2}) \hat{i} + (z^{2} + x^{2} - y^{2}) \hat{j} + (x^{2} + y^{2} - z^{2}) \hat{k}$ .

[2009, 20M]

9) Determine $\int_{c} (y d x + z d y + x d z)$ by using Stoke’s theorem, where $C$ is the curve defined by $(x - a)^{2} + (y - a)^{2} + z^{2} = 2 a^{2}$ , $x + y = 2 a$ that starts from the point $(2 a, 0, 0)$ goes at first below the $z - p l a n e$ .

[2007, 15M]

10) Verify Stokes’ theorem for the function $\bar{F} = x^{2} \hat{i} - x y \hat{j}$ integrated round the square in the plane $z = 0$ and bounded by the lines $x = 0$ , $y = 0$ , $x = a$ and $y = a, a > 0$ .

[2006, 15M]

11) Verify Stokes’ theorem for $\hat{f} = (2 x - y) \hat{i} - y z^{2} \hat{j} z \hat{k}$ where $S$ is the upper half surface of the sphere $x^{2} + y^{2} + z^{2} = 1$ and $C$ is its boundary.

[2004, 15M]

12) Evaluate $\iint_{S} c u r l A . d S$ , where $S$ is the open surface

x^{2} + y^{2} - 4 x + 4 z = 0, z \geq 0

and

$A = (y^{2} + z^{2} - x^{2}) \hat{i}$ + $(2 z^{2} + x^{2} - y^{2}) \hat{j}$ + $(x^{2} + y^{2} - 3 z^{2}) \hat{k}$

[2003, 15M]

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