3FI4: Theory and Applications in Electromagnetics Lecture 2: Vector Analysis: Differential Calculus page 1 September 12, 2000 1. Scalar Fields and Vector Fields The simplest possible physical field is the scalar field. It represents a function depending on the position in space (and time). A scalar field is characterized at each point in space (and time) by a single number. Examples of scalar fields: temperature, gravitational potential, electrostatic potential (voltage), etc.
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1. Scalar Fields and Vector Fields The simplest possible ...hep.fcfm.buap.mx/cursos/2002/CV/Lecture_2.pdf · A scalar field is characterized at each point in space ... the corresponding
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The simplest possible physical field is the scalar field. Itrepresents a function depending on the position in space (andtime). A scalar field is characterized at each point in space(and time) by a single number.
Examples of scalar fields: temperature, gravitationalpotential, electrostatic potential (voltage), etc.
The gradient of a scalar field is a vector whose magnitudeis equal to the maximum rate of change of the field andwhose direction is equal to the direction of the fastestincrease.
Assume a scalar field ( , , )x y zΦAn infinitesimal displacement along the x-axis dx brings usto a slightly different scalar value ( , , )x dx y zΦ +
ˆ( , , ) ( , , )xd x dx y z x y z dl xdxΦ = Φ + − Φ =ˆ( , , ) ( , , )yd x y dy z x y z dl ydyΦ = Φ + − Φ =ˆ( , , ) ( , , )zd x y z dz x y z dl zdzΦ = Φ + − Φ =
is a vector showing the direction of spatial displacement andthe corresponding rate of change of the scalar function.
In orthogonal coordinate system, the gradient is a vectorialsum of the directional derivatives of the scalar field alongthe three unit vectors of the system
To estimate and quantify vector fields, it is often useful tomeasure their flow (or net outflow). Such a measure isthe vector field flux. The flux is the net normal flowthrough a surface
cosS S
F ds F dsθΨ = ⋅ =∫∫ ∫∫
ˆds nds=Differential surface area (surface element):
The flux (or the divergence) is not enough to describe avector field. It is descriptive only with regard to thecomponents of the field normal to a given surface. Thefield components tangential to a surface (or a contour of asurface) are important, too. Thus, a vector field is alsocharacterized by its circulation.
The curl vector has three orthogonal components in anorthogonal coordinate system. In the RCS, the circulation inthe x-y plane defines the z-component of the curl; thecirculation in the y-z plane defines the x-component; and thecirculation in the x-z plane defines the y-component of thecurl.
The net flux of the curl of a vector over any open surface Sis equal to the line integral of the vector along the closedcontour C enclosing the surface.
[ ]
( )SC S
F dl F ds⋅ = ∇ × ⋅∫ ∫∫
Adding up the individual circulationsof all sub-cells results in a netcirculation over the contour of thewhole surface.