ElectroMagnetic Field Equations
Coordinate Systems: A point P is described by three coordinates , in cartesian (x,y,z), in cylindrical (r, ϕ ,z), and in spherical (r,θ, ϕ ), as shown.
A point is also defined by the intersection of three orthogonal surfaces, as shown.
Cartesian Cylindrical Spherical Transformation from Cylinder to Cartesian Coordinates
Transformation from Cartesian to Cylindrical:
Transformation from Spherical to Cartesian:
The inverse transformation
Differential Lengths, Surfaces and Volumes
When integrating along lines, over surfaces, or throughout volumes, the ranges of the respective variables define the limits of the respective integrations. In order to evaluate these integrals, we must properly define the differential elements of length, surface and volume in the coordinate system of interest. The definition of the proper differential elements of length (dl for line integrals) and area (ds for surface integrals) can be determined directly from the definition of the differential volume (dv for volume integrals) in a particular coordinate system.
Rectangular Coordinates
Cylindrical Coordinates
Spherical Coordinates
Line Integrals of Vectors
The component of a vector along a given path is found using the dot product. The resulting scalar function is integrated along the path to obtain the desired result. The line integral of the vector A along a the path L is then defined as shown in fig.
Where dl = al dl al : unit vector in the direction of the path L dl : differential element of length along the path L
A.dl = A.al dl = Al dl
Al : component of A along the path L
Whenever the path L is a closed path, the resulting line integral of A is defined as the circulation of A around L and written as
Example (Line integral)
Given H=(x-y)ax +(x2+zy)ay +5yz az, evaluate the line integral of H along the path L made
A point is also defined by the intersection of three orthogonal surfaces, as shown.
Cartesian Cylindrical Spherical Transformation from Cylinder to Cartesian Coordinates
Transformation from Cartesian to Cylindrical:
Transformation from Spherical to Cartesian:
The inverse transformation
Differential Lengths, Surfaces and Volumes
When integrating along lines, over surfaces, or throughout volumes, the ranges of the respective variables define the limits of the respective integrations. In order to evaluate these integrals, we must properly define the differential elements of length, surface and volume in the coordinate system of interest. The definition of the proper differential elements of length (dl for line integrals) and area (ds for surface integrals) can be determined directly from the definition of the differential volume (dv for volume integrals) in a particular coordinate system.
Rectangular Coordinates
Cylindrical Coordinates
Spherical Coordinates
Line Integrals of Vectors
The component of a vector along a given path is found using the dot product. The resulting scalar function is integrated along the path to obtain the desired result. The line integral of the vector A along a the path L is then defined as shown in fig.
Where dl = al dl al : unit vector in the direction of the path L dl : differential element of length along the path L
A.dl = A.al dl = Al dl
Al : component of A along the path L
Whenever the path L is a closed path, the resulting line integral of A is defined as the circulation of A around L and written as
Example (Line integral)
Given H=(x-y)ax +(x2+zy)ay +5yz az, evaluate the line integral of H along the path L made