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Kalkulus Vektor
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OPERATOR OPERATOR DIFERENSIAL VEKTORDIFERENSIAL VEKTOR
Pendahuluan 0
0
0
Medan Skalar0
Vektor gradien
0
= ∇ = + +
0
a) ∇ ( + ) = ∇ + ∇ b) ∇ ( ) = ∇ c) ∇ ( ) = (∇ ) + (∇ )
Bukti a)
∇ ( + ) = ( + )
+ ( + )
+ ( + )
= + + + + +
= + + + + +
= ∇ + ∇
Bukti b)
terbukti ∇ ( + ) = ∇ + ∇
∇ ( ) = + +
= + +
= + +
= ∇ Terbukti ∇( ) = ∇
0
0
Medan vektor0
Divergensi vektor0
= ∇ ∙ = 1( , , )
+ 2( , , )
+ 3( , , )
Soal 1.cjelaskan bahwa
Jawab :
= ∇ ∙ = 1( , , )
+ 2( , , )
+ 3( , , )
= ∇ ∙ = + + ( 1 + 2 + 3 )
= ∙ 1 + ∙ 2 + ∙ 3
= 1 + 2 + 3
Rotasi vektor
0
= ∇ ×
=
1 2 3
Jawab :
= ∇ × =
1 2 3
= 3 − 2 − 3 − 1 + 2 − 1
Soal nomor 2
Diketahui fungsi vektor
Dan fungsi skalar
Tentukanlah:
( , , ) = (2 2 ) + ( ) + (3 3) ( , , ) = (3 3) + (2 ) − ( 2)
f(x,y,z) = xy – x2yz
) ∇ ( ∙ ∇ × )
b) ∇ ∙ ∇
c) ( × ∇) × pada titik (2,-1,3)
d) ∇ ∙ [∇ × ]
0
0
