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Summary

  • A quantity that has only magnitude is said to be a scalar quantity, such as time, mass, distance, temperature, work, electric potential, etc. A quantity that has both magnitude and direction is called a vector quantity, such as force, velocity, displacement, electric field intensity, etc.
  • If the value of the physical function at each point is a scalar quantity then the field is known as a scalar field, such as temperature distribution in a building. If the value of the physical function at each point is a vector quantity then the field is known as a vector field, such as the gravitational force on a body in space.
  • A unit vector Description: 81254.png along Description: 81243.png is defined as a vector whose magnitude is unity and its direction is along Description: 81236.png In general, any vector can be represented as
Description: 81229.png 
where A or Description: 81220.png represents the magnitude of the vector and Description: 81211.png, direction of the vector Description: 81204.png.
  • Two vectors can be added together by the triangle rule or parallelogram rule of vector addition.
  • The dot product of two vectors Description: 81196.png and Description: 81187.png, written as Description: 81180.png, is defined as,
Description: 81173.png
where Description: 81163.png is the smaller angle between Description: 81152.png and Description: 81145.png, and Description: 81138.png and Description: 81129.png represent the magnitude of Description: 81121.png and Description: 81114.png, respectively.
  • The cross product of two vectors Description: 81107.png and Description: 81097.png, written as Description: 81090.png, is defined as,
Description: 81083.png 
where Description: 81073.png is the unit vector normal to the plane containing Description: 81061.png and Description: 81054.png. The direction of the cross product is obtained from a common rule, called right-hand rule.
  • ​Three orthogonal coordinate systems commonly used are Cartesian coordinates (x, y, z), cylindrical coordinates (r, φ, z) and spherical coordinates (ρθφ).
  • The differential lengths in three coordinate systems are given respectively as,
Description: 81046.png
 
Description: 81038.png
 
Description: 81030.png
  • The differential areas in three coordinate systems are given respectively as,
Description: 81023.png
 
Description: 81016.png
 
Description: 81005.png
  • The differential volumes in three coordinate systems are given respectively as,
Description: 80998.png
 
Description: 80991.png
 
Description: 80977.png
  • For the vector Description: 80966.png and a path l, the line integral is given by,
Description: 80958.png
  • If the path of integration is a closed curve, the line integral is the circulation of the vector around the path.
  • If the line integration of a vector along a closed path is zero, i.e., Description: 101733.png then the vector is known as conservative or lamellar.
  • For a vector Description: 80943.png, continuous in a region containing a smooth surface S, the surface integral or the flux of Description: 80935.png through S is defined as,
Description: 80927.png
where, Description: 80920.png is the unit normal vector to the surface S.
  • If the surface is a closed surface, the surface integral is the net outward flux of the vector.
  • If the surface integral of a vector over a closed surface is zero, i.e., Description: 80911.png then the vector is known as a solenoidal vector.
  • The volume integral of a scalar quantity F over a volume V is written as,
Description: 80904.png
  • The differential vector operator () or del or nabla, defined in Cartesian coordinates as,
​​Description: 80896.png 
is merely a vector operator, but not a vector quantity. It is a vector space function operator. It is used for performing vector differentiations.
  • The gradient of a scalar function is both the magnitude and the direction of the maximum space rate of change of that function.
  • The gradient of a scalar quantity in three different coordinate systems is expressed respectively as,
Description: 80885.png
  • The divergence of a three-dimensional vector field at a point is a measure of how much the vector diverges or converges from that point.
  • If the divergence of a vector is zero then the vector is known as a solenoidal vector.
  • The divergence of a vector quantity in three different coordinate systems is expressed respectively as,
Description: 80878.png
  • The curl of a vector field, denoted by curl Description: 80862.png or Description: 80855.png, is defined as the vector field having magnitude equal to the maximum circulation at each point and to be oriented perpendicularly to this plane of circulation for each point.
  • If the curl of a vector is zero then the vector is known as irrotational vector.
  • The curl of a vector quantity in three different coordinate systems is expressed respectively as,
Description: 80846.png
  • The Laplacian operator (2) of a scalar field is the divergence of the gradient of the scalar field upon which the operator operates.
  • The Laplacian operator (2) of a scalar field in three different coordinate systems is expressed respectively as,
Description: 80839.png
  • Gauss’ divergence theorem is used to convert a volume integral into surface integral and vice versa. According to this theorem, the divergence of a vector field over a volume is equal to the surface integral of the normal component of the vector through the closed surface bounding the volume.
Description: 80816.png
where V is the volume enclosed by the closed surface S.
  • According Green’s theorem, if Description: 80808.png is a two-dimensional vector field, such that Description: 80798.pngthen,
Description: 80788.png 
where C is a positively oriented boundary of the region R.
  • Stokes’ theorem is used to convert a line integral into surface integral and vice versa. According to this theorem, the line integral of a vector around a closed path is equal to the surface integral of the normal component of is curl over the surface bounded by the path.
Description: 80779.png
where S is the surface enclosed by the path L. The positive direction of Description: 80771.png is related to the positive sense of defining L according to the right-hand rule.
  • The Helmholtz theorem states that a vector field is uniquely described within a region by its divergence and curl.




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