Engineering Mathematics: Vector Analysis

By Yash Bansal|Updated : June 8th, 2021

 

 

 

 

 

 

 

                                                                     

 

 

 

 

Introduction:

Principal application of vector function is the analysis of motion is space. The gradient defines the normal to the tangent plane, the directional derivatives give the rate of change in any given direction. If  is the velocity field of a fluid flow, then divergence of E at  appoint P (x, y, z) (Flux density) is the rate at which fluid is (diverging) piped in or drained away at P, and the curl  (or circular density) is the vector of greatest circulation in flow, we express grad, div and curl in general curvilinear. Coordinate and in cylindrical and spherical. Coordinates which are useful in engineering physics or geometry involving a cylinder or cone or a sphere.

Coordinate System:

1. Cartesian Coordinate System.

In the Cartesian system, the 3 base vectors are Any point in space can be written in the form,  

where (x1, y1, z1) are the coordinates of the point P in the Cartesian space which is the intersection of the three planes x = x1, y = y1, z = z1. The distance of the point from the origin is given by,The figure below depicts the point P in the Cartesian Coordinate System. As you can see x1,y1 and z1 can also be understood as the perpendicular distance of point P from the YZ, XZ and XY plane,

 

  • The differential length, differential surface and differential volume are given by,

     

2. Cylindrical Coordinate System.

In the cylindrical system, the three base vectors are. Any point in space can be written in the form, ,

where (r1, φ1, z1) are the coordinates of the point P in the Cylindrical Space. The point P is the intersection of a circular cylinder surface r = r1, a half plane containing z-axis and making an angle φ = φ1 with the XZ plane and a plane z = z1 parallel to the XY plane. φ1 is measured from the positive x-axis and the base vector ˆaφ is tangential to the cylindrical surface.

 

  • The distance of the point from the origin is given by, 
  • The values of the 3 coordinates vary as follows,

        

  • the differential length, differential surface and differential volume are given by, 

     

3. Spherical Coordinate System.

In the Spherical system, the 3 bases are ˆar, ˆaθ and ˆaφ. Any point in space can be written in the form, 

where (r, θ, φ) are the coordinates of the point P in the Spherical Space. A point P(r1, θ1, φ1) in the spherical coordinates are specified as the intersection of the following three surfaces: a spherical surface centered at origin and has a radius r1, a right circular cone with its apex at origin and half angle θ1 and a half plane containing z-axis and making an angle φ1 with the XZ plane,

  • The distance of the point from the origin is given by, 
  • The values of the 3 coordinates vary as follows, 

       

  • the differential length, differential surface and differential volume are given by,

      

 Scalar and Vector Products:

  • Dot Product: is also called scalar product. Let ‘θ’ be the angle between vectors A and B. 

                                       

  • Cross product: is also called vector product.                                  

              

       S = |S| ân where |S| = |A| |B| Sinθ, To find the direction of S, consider a right threaded screw being rotated from A to B. i.e. perpendicular to the plane containing the vectors A and B. therefore, 

Cartesian coordinate to Cylindrical coordinate Conversion:

Point transformation,

The relationship between  are vector transformation,

Cartesian coordinate to Spherical coordinate:

Point transformation,

 

 

 Gradient, Curl & Divergence:

  • Gradient: Gradient of a scalar field V, is a vector that represents both magnitude and direction of maximum space rate of change of V. The gradient of V, ∇V, will always be perpendicular to a constant V surface. If  = ∇V, then V is said to be the scalar potential of .

The vector that represents the magnitude and direction of the maximum space rate of increase of a scalar is the gradient of that scalar. 

It can be shown that the gradient operator in the three coordinate system are 

 

  • A divergence of a Vector Field: The spatial derivatives of a vector field are represented through divergence and curl. It is usually convenient to represent vector field variations in space as field lines or flux lines whose directions indicate the direction of these lines.

  • Divergence of a vector field at a given point P is the outward flux per unit volume, as the volume shrinks about P. Divergence of a vector field is a scalar 
  • The divergence operator in the 3 coordinate systems:

        

  • Divergence theorem relates the divergence of a vector field to the surface integral of over a surface. It is given by where S is the surface and V in the volume enclosed by the surface S.
  • Curl: Similar to a flow source, vector fields can also exist as ’vortex sources’ which causes circulation of a vector field around it, Ifis a force acting on an object, circulation would be the work done by the force in moving the object once around the contour. The curl of is a vector whose magnitude is the maximum circulation of per unit area, as the area tends to zero. The direction of the curl is the normal direction of the area when the area is oriented to make the circulation maximum.

        

  • Stokes Theorem relates the curl of a vector field to the line integral of over a contour C. It is given by, 
  • Given below is the ∇ operator in the 3 coordinate systems

       

 Laplacian Operation:

The Laplacian of a scalar field V in different coordinate systems is defined as

  • Laplacian of a Vector function ():

      

                       

Thanks 
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Posted by:

Yash BansalYash BansalMember since Nov 2018
Content Manager (GATE)
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