Vector Analysis
Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation
and integration of vector fields.
and integration of vector fields.
Differential length, Area and Volume in different coordinate systems
Conversion from spherical and cylindrical to Cartesian coordinate system
·
Gradient
of a scalar field:
The vector that represents both the magnitude and the direction
of the maximum space rate of increase of a scalar as the gradient of that
scalar.
Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.
Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.
·
Divergence
of the vector field: the divergence of a
vector field A at a point, abbreviated div A, as the net outward flux of A per
unit volume as the volume about the point tends to zero:
Measures the scalar of a source or sink at a given point in a vector field. Maps vector fields to scalar fields.
· Curl of a vector field: The curl of u vector field A, denoted by Curl A , is a vector whose magnitude is the maximum net circulation of A per unit urea as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum.
Measures the tendency to rotate about a point in a vector field. Maps vector fields to (pseudo)vector fields.
Measures the scalar of a source or sink at a given point in a vector field. Maps vector fields to scalar fields.
· Curl of a vector field: The curl of u vector field A, denoted by Curl A , is a vector whose magnitude is the maximum net circulation of A per unit urea as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the net circulation maximum.
Measures the tendency to rotate about a point in a vector field. Maps vector fields to (pseudo)vector fields.
·
Divergence
theorem: The volume integral of divergence of a
vector field equals the total outward flux of the vector through the surface
that bounds the volume;
·
Stoke
theorem: The surface integral of the curl of
vector field over an open surface is equal to the closed line integral of the
vector along the contour bounding the surface.
Electrostatics and magnetostatics:
·
Coulomb’s Law: The
electric field intensity of a point charge is in the outward radial direction
and has a magnitude proportional to the charge and inversely proportional to
the square of the distance from the charge.
·
Gauss’s
Law: The net electric flux through
any closed surface is equal
to 1⁄ε times the net electric charge enclosed
within that closed surface.
Formulas used in Electrostatics and magnetostatics are as follows:
Time Varying Fields and Maxwell’s Equations:
