Divergence and curl of vectorfields


PEar tree 'Gris Bonne'

Pyrus Communis (pear tree) ‘Gris Bonne’

According to the Helmholtz-theorem a vectorfield is completely defined by the divergence and  curl of the vectorfield.

the divergence is a measure of the strength of the source of the vectorfield whereas the degree of rotation of the field is given by the curl.

The divergence is defined as  ·F  = lim Δv→0 ∫A ds/Δv i.e. the scalarproduct(dotproduct) of the nabla operator and the vector.

The ∇-operator is defined as the vector differential operator
∇=∂/∂x + ∂/∂y + ∂/∂z.

When this operates on a scalar V one obtains the gradient  V of that scalar i.e. a vector that represents both the magnitude and the direction of the maximum space rate of increase of  of that scalar.

The curl is defined by

∇xF. = (dFz/dy – dFy/dz) i + (dFx/dz – dFz/dx)j + (dFy/dx – dFx/dy) k

The electromagnetic field is defined by the divergence and curl of the Electric field vector E and the magnetic field vector B:
∇· E= ρ

∇xE=∂B/∂t

∇· B=0; This can be interpretated as stating the fact that there are no magnetic charges.
∇xB=∂D/∂t

These are the famous Maxwellian equations which gives a full description of the electromagnetic theory.
Every electromagnetic law can be deduced from them.

Annonser

Om mattelararen

Licentiate of Philosophy in atomic Physics Master of Science in Physics
Det här inlägget postades i Calculus, Uncategorized, Vectors. Bokmärk permalänken.

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