Apr 19, 2023 G L Pollack and D R Stump Electromagnetism

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2. Vector Calculus Chapter Summary

• The grad, div, and curl in Cartesian coordinates are

zyx

zyx

FFF

zyx

z

F

y

F

x

F

z

f

y

f

x

ff

///

ˆˆˆ

ˆˆˆ

kji

F

F

kji

Apr 19, 2023 G L Pollack and D R Stump Electromagnetism

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• The coordinate-independent definitions of grad, div, and curl are

lFF

AFF

x

ddA

ddV

dfdf

1lim

1 lim

§ In the second equation, dV is a small volume and the limit means dV shrinks to a point. The integral is the flux of F outward through the boundary of dV. The divergence is a scalar.§ In the third equation, dA is a small surface area which shrinks to a point. The integral is the line integral around the boundary of dA. The curl is a vector, so it is implied by the equation that the right-hand side is the component of F normal to the surface.)

Apr 19, 2023 G L Pollack and D R Stump Electromagnetism

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• The grad, div, and curl in cylindrical or spherical coordinates are not simple generalizations of the Cartesian equations, because of scale factors. The formulas for these operators are in Tables 2.3 and 2.4.

• Two important integral identities are

CS

SV

3

theoremsStokes'

theoremsGauss'

lFAF

AFF

dd

dxd