UNIT - I VECTOR...
Transcript of UNIT - I VECTOR...
![Page 1: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/1.jpg)
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UNIT - I
VECTOR CALCULUS
Definition:
Definition: Gradient of a scalar function
Let ( ) be a scalar point function and is continuously differentiable then the vector
(
)
is called gradient of the function and is denoted as .
Example:
If , find at ( )
Answer:
( ) ( ) ( )
( ) ( ) ( )
( )
Example:
If , find
Answer:
![Page 2: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/2.jpg)
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Example:
If Find | | at ( )
Answer:
( ) ( ) ( )
( )
| | √ √
Example:
Find the gradient of the function ( ) .
Answer:
( )
[ ( ) ( ) ( )]
![Page 3: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/3.jpg)
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Example:
Find where .
Answer:
Differentiating partially with respect to respectively
[ ]
Example:
Find where .
Answer:
| | √
√
√
√
√
√ [ ( ) ( ) ( )]
√
![Page 4: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/4.jpg)
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Definition: Unit normal to the surface
| |
Example:
Find the unit vector normal to the surface ( ).
Answer:
( )
| | √ √ √
| |
√
√
Example:
Find the unit vector normal to the surface ( ).
Answer:
( ) ( ) ( )
( )
| | √ √
| |
√
Definition: Angle between the surfaces
The angle between the surfaces is given by
| || |
Example:
![Page 5: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/5.jpg)
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Find the angle between the surfaces ( )
Answer:
( )
| | √ √
( )
| | √ √
| || |
( ) ( )
√ √
√
√
Example:
Find the angle between the surfaces ( )
Answer:
( )
| | √ √
( )
| | √
![Page 6: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/6.jpg)
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| || |
( ) ( )
Definition: Divergence of a scalar point function
If ( ) is a continuously differentiable vector point function in a region of space then the
divergence is defined by
( ) (
) ( )
Definition: Curl of a vector function
If ( ) is a continuously differentiable vector point function defined in each point
( ) then the is defined by
||
|| (
) (
) (
)
Example:
If , then find .
Answer:
![Page 7: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/7.jpg)
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( ) ( ) ( )
( )
||
|| ||
||
(
) (
) (
)
Example:
If ( ) , then find at ( ).
Answer:
( )
( ) ( ) ( )
[ ]( )
( )
||
|| ||
||
(
( )
) (
( )
) (
)
( ) ( ) ( )
[ ]( )
Definition: Laplace Equation
![Page 8: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/8.jpg)
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Example:
(
)
Answer:
∑
∑
(
( ) )
∑
( )
( )
( )
( )
( )
Definition: Solenoidal vector
Definition: Irrotational vector
Definition: Scalar Potential
Such a scalar
function is called scalar potential of .
Example:
Prove that is Solenoidal.
Answer:
![Page 9: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/9.jpg)
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Hence is Solenoidal.
Example:
Prove that is Solenoidal.
Answer:
( )
( )
( )
Hence is Solenoidal.
Example:
If ( ) ( ) ( ) is Solenoidal find the value of .
Answer:
( ) ( ) ( )
( )
( )
( )
Example:
Prove that is Irrotational.
Answer:
![Page 10: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/10.jpg)
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||
|| |
|
(
) (
) (
)
( ) ( ) ( )
Hence is Irrotational.
Example:
Find the constants so that ( ) ( ) ( ) is
Irrotational.
Answer:
( ) ( ) ( )
||
|| ||
||
[
( )
( )]
[
( )
( )]
[
( )
( )]
( ) ( ) ( )
Example:
![Page 11: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/11.jpg)
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Prove that ( ) ( ) ( ) is Solenoidal as well as Irrotational. Also find
the scalar potential of .
Answer:
( ) ( ) ( )
( )
( )
( )
Hence is Solenoidal.
( ) ( ) ( )
||
|| ||
||
(
( )
( ))
(
( )
( ))
(
( )
( ))
( ) ( ) ( )
Hence is Irrotational.
To find scalar potential:
To find such that
![Page 12: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/12.jpg)
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( ) ( ) ( )
Integrating with respect to respectively, we get
( ) ( )
( ) ( )
( ) ( )
Combining, we get ( )
where is a constant.
Therefore is a scalar potential.
Example:
Prove that ( ) ( ) ( ) is Irrotational. Also find the scalar
potential of .
Answer:
( ) ( ) ( )
||
|| ||
||
(
( )
( ))
(
( )
( ))
![Page 13: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/13.jpg)
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(
( )
( ))
( ) ( ) ( )
Hence is Irrotational.
To find scalar potential:
To find such that
( ) ( ) ( )
Integrating with respect to respectively, we get
( ) ( )
( ) ( )
( ) ( )
( )
where is a constant. Therefore is a scalar potential.
Vector integration
Line integral
Let be a vector field in space and let be a curve described in the sense .
∑
∫
If the line integral along the curve then it is denoted by
![Page 14: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/14.jpg)
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∫
∮
Example:
If ( ) . Evaluate ∮
from ( ) ( ) along the
curve.
Solution:
The end points are ( ) ( )
These points correspond to
∮
∫[( ) ]
∫[( ) ]
∫[ ]
*
+
Example:
Find the work done by the moving particle in the force field ( )
from along the curve
Solution:
∮
∫[ ( ) ]
∫[ [ ] ]
![Page 15: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/15.jpg)
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∫[ ]
*
+
Greens theorem in plane
If ( ) ( ) are continuous functions with continuous, partial derivatives in a region of
the plane bounded by a simple closed curve then
∫
∬(
)
where is the curve described in the positive direction.
Example:
Evaluate using Green’s theorem in the plane for ∫( ) where is the closed curve of
the region bounded by and .
Answer:
∫
∬(
)
∫ ( )
∬ ( )
∬( )
∫ ∫ ( )
√
∫ *
+
√
(0,0)
(1,1)
![Page 16: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/16.jpg)
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∫ *(
√ ) (
)+
∫ [(
)]
(
)
(
)
Example:
Verify Green’s theorem in the plane for ∫ ( )
where is the curve in the plane given by
( ).
Answer:
∫
∬(
)
∫
∫
∫
∫
∫
Along
∫
∫
*
+
Along
∫
∫
*
+
Along
∫
∫
*
+
Along
( )
( )
( )
( )
![Page 17: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/17.jpg)
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∫
∫
∫
∬(
)
∫∫
∫ [ ]
*
+
∫
∬(
)
Example:
Verify Green’s theorem in the plane for ∫ ( ) ( )
where is the boundary
of the region bounded by
Answer:
∫
∬(
)
∬ (
)
∫ ∫
∫ [ ]
∫ ( )
∫[ ]
(
)
(
)
( )
( )
( )
![Page 18: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/18.jpg)
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∫( ) ( )
∫
∫
∫
Along
∫( )
∫( )
*
+
Along
∫( ) ( )
∫( ( ) ) ( ( ) ( ))( )
∫( )
*
+
Along
∫( ) ( )
∫
∫
![Page 19: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/19.jpg)
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∫
∫
∬(
)
Example:
Verify Green’s theorem in the plane for ∫ ( ) ( )
where is the square
bounded by
Answer:
∫
∬(
)
∫ ( ) ( )
∫
∫
∫
∫
Along
∫ ( )
( ) *
+
( )
Along
∫( )
*
+
Along CD
∫ ( )
( ) *
+
( ) (
)
( )
( )
( ) ( )
![Page 20: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/20.jpg)
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Along DA
∫ ( )
*
+
∫
( )
∬(
)
∫ ∫( )
∫ *
+
∫
[ ]
( )
∫
∬(
)
Example:
Find the area bounded between the curves using Green’s theorem.
Answer:
∫
∬(
)
The point of intersection of are ( ) ( ).
The area enclosed by a single curve is
∫ ( )
.
∫( )
∫( )
Along ( )
![Page 21: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/21.jpg)
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∫( )
∫(
)
∫
*
+
Along ( )
∫( )
∫(
)
∫
*
+
(
)
Stoke’s theorem
If is a open surface bounded by a simple closed curve and if a vector function is continuous and
has continuous partial derivatives in and so on , then
∫
∬
where is the unit vector normal to the surface. That is, the surface integral of the normal component
of is equal to the line integral of the tangential component of taken around .
Example:
Verify Stoke’s theorem for a vector field defined by ( ) in the rectangular region in
the plane bounded by the lines
Answer:
∫
∬
( )
![Page 22: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/22.jpg)
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( )
||
|| ||
||
(
( )
( )) (
( )
( )) (
( )
( ))
( ) ( ) ( )
Here the surface denotes the rectangle and the unit outward normal to the vector is .
That is
∬
∬
∬
∬
∫∫
∫ [ ]
*
+
∫
∫( )
∫
∫
∫
∫
Along
∫
*
+
Along
∫
*
+
( )
( )
( )
( )
![Page 23: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/23.jpg)
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Along
∫( )
*
+
Along
∫( )
∫
∫
∬
Example:
Verify Stoke’s theorem when ( ) ( ) and is the boundary of the region
enclosed by the parabolas
Answer:
∫
∬
( ) ( )
( ) ( )
||
|| ||
||
![Page 24: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/24.jpg)
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(
( )
( )) (
( )
( ))
(
( )
( ))
( ) ( ) ( )
Here the surface denotes the surface of the plane. The unit outward normal to the vector is .
That is
∬
∬
∬
∬
∫ ∫
√
∫ *
+
√
∫( )
*
+
(
)
∫
∫
∫( ) ( )
∫
∫
Along
∫( ) ( )
![Page 25: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/25.jpg)
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∫( )
∫( )
*
+
(
)
Along
∫( ) ( )
∫( )
∫( )
*
+
(
)
∫
∫
∬
Example:
Verify Stoke’s theorem for over the upper half of the sphere
.
Answer:
∫
∬
O(0,0)
( )
![Page 26: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/26.jpg)
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[ ]
||
|| ||
||
(
( )
( ) ) (
( )
( )) (
( )
( ))
( ) ( ) ( )
| | √
| |
Here the surface denotes the rectangle and the unit outward normal to the vector is .
( ) (
)
( )
∬
∬
| |
∬(
)
![Page 27: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/27.jpg)
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∫ ∫ ( )
√
√
∫ *
+ √
√
∫[ ]
∬
∫
∫
∫
∫
∫
∫
Since
∫
∫ [ ( ) ( )]
∫
∫
∫ ( )
[
]
[
]
[
]
[
]
[
]
∫
∬
Example:
![Page 28: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/28.jpg)
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Verify Stoke’s theorem for the function integrated round the square in the plane
whose sides are along the line .
Answer:
∫
∬
||
|| ||
||
(
( )
( )) (
( )
( )) (
( )
( ))
( ) ( ) ( )
Here the surface denotes the rectangle and the unit outward normal to the vector is .
That is
∬
∬
∬
∬
∫ ∫
∫
∫
∫
∫
∫
∫
![Page 29: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/29.jpg)
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Along
∫
∫
*
+
Along
∫
∫
*
+
Along
∫
∫
*
+
Along
∫
∫
∫
∫
∫
∬
Example:
Verify Stoke’s theorem for the function where is the open surface of the
cube formed by the planes in which the plane is cut.
Answer:
∫
∬( )
( )
( )
( )
( )
![Page 30: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/30.jpg)
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||
|| ||
||
(
( )
( ))
(
( )
( ))
(
( )
( ))
( ) ( )
( )
Since the surface is cut.
∬( )
∬ ∬ ∬ ∬ ∬
Surface Equation ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
∬
∬( ) ∬( )
∬( ) ∬( ) ∬( )
∬ ∬ ∬( )
![Page 31: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/31.jpg)
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∫ ∫
∫ ∫
∫ ∫( )
∫[ ]
∫[ ]
∫[ ]
*
+
∬
∫
∫
∫
∫
∫
The boundary bounded by the square that lies in the plane .
Along AB
∫
[ ]
Along
∫
[ ]
Along
∫
Along
( )
( )
( ) ( )
![Page 32: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/32.jpg)
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∫
∫
∫
∬
Gauss Divergence theorem
Statement:
The surface integral of the normal component of a vector function over a closed surface enclosing
the volume is equal to the volume of integral of the divergence of taken throughout the volume .
∬
∭
Example:
Verify Gauss divergence theorem for over the cube bounded by
Answer:
∬
∭
∭
∫∫∫( )
∫∫[ ]
∫∫[ ]
∫*
+
∫(
)
(
)
![Page 33: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/33.jpg)
33
∬
∬ ∬ ∬ ∬ ∬ ∬
Surface Equation
( ) ( ) ( ) ( ) ( )
( )
∬
∫∫
∫∫
∫∫
∬
Gauss divergence theorem is verified.
Example:
Verify Gauss divergence theorem for ( ) ( ) ( ) over the
rectangular parallelepiped
.
Answer:
∬
∭
( )
∭
![Page 34: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/34.jpg)
34
∫∫∫ ( )
∫∫*
+
∫∫*
+
∫ *
+
∫(
)
*
+
∭
( )
∬
∬ ∬ ∬ ∬ ∬ ∬
Surface Equation
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
∬
∫∫( )
∫(
)
∫(
)
(
)
∬
∫∫ ( )
∫(
)
∫(
)
![Page 35: UNIT - I VECTOR CALCULUStranquileducation.weebly.com/uploads/1/3/7/6/13765138/vector_calculus.pdf · where ̂ is the unit vector normal to the surface. That is, the surface integral](https://reader033.fdocuments.us/reader033/viewer/2022041804/5e5394224e6eb65b9d18b460/html5/thumbnails/35.jpg)
35
(
)
∬
∫∫( )
∫(
)
∫(
)
(
)
∬
∫∫ ( )
∫(
)
∫(
)
(
)
∬
∫∫( )
∫(
)
∫(
)
(
)
∬
∫∫ ( )
∫(
)
∫(
)
(
)
∬
( )
Gauss divergence theorem is verified.