Chapter: Mathematics (maths) - Vector Calculus

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Vector Calculus

1 Gradient-Directional Derivative 2. Divergence And Curl –Irrotational And Solenoidal Vector Fields Divergence 3 Vector Integration 4 Green’s Theorem In A Plane;(Excluding proof) 5 Gauss Divergence Theorem:(Excluding proof) 6 Stoke’s Theorem(Excluding proof)

VECTOR CALCULUS

 

1        Gradient-Directional Derivative

2.       Divergence And Curl –Irrotational And Solenoidal Vector Fields Divergence 

3        Vector Integration

4        Green’s   Theorem   In   A   Plane;(Excluding proof)

5        Gauss Divergence Theorem:(Excluding proof)

6        Stoke’s   Theorem(Excluding   proof)

 

 

 

VECTOR CALCULUS

 

1Gradient-Directional Derivative

 

1.1. Gradient

1(a) The Vector Differential Operator



1(b) The Gradient (Or Slope Of A Scalar Point Function)




1.2.  Directional Derivative



1.3. Unit Tangent Vector


1.Find  a unit tangent vector to the following surfaces at the specified


1.4 Normal Derivative



1.5 Unit Normal Vector



1.6 Angle Between The Suraces




1.7 .Scalar Potential 



1. 8 The Vector Equation of the Tangent Plane And Normal Line to the Surface







Tutorial Problems:


 

 

2 Divergence And Curl Irotational And Solenoidal Vector Fields:

2.1 Divergence and curl





2.2   SOLENOIDAL VECTOR,IRROTATIONAL VECTOR:

Solenoidal vector formula:






Laplace Operator:


 

3 Vector Integration

 

Conservative Vector Field:



3.1. Line Integral:





3.2. Surface Integral:


Definition: Consider a surface S .Let n denote the unit outward normal to the surface S. Let R be the projection of the surface x on xy plane. Let Vec f be a vector function defined in some region  containing the surface S, then the surface integral of Vector f is defined to be 





3.3. Volume Integral:



3.4 Tutorial Problems:



 

4 .Green’s Theorem In A Plane:

Statement:


 

Tutorial Problems:



 

5 Gauss Divergence Theorem:

Statement:

          The surface integral of the normal component of a vector function F over a closed surface  S enclosing volume V is equal to the volume integral of the divergence of F taken throughout the


 



6. Stoke’s Theorem

Statement:

The surface integral of the normal component of the curl of a vector function F over an open surface S is equal to the line integral of the tangential component of F around the closed curve C bounding S.      



Hence, Stoke’s  theorem is verified.



6.1 Tutorial Problems:




 


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