Question

3. Divergence Find the divergence of: a) Ē(x, y, z)=(-2y x 0] b) F(x,y,z)= (y2 – 2x 5x’y x+37] c) v =[3y–2yx xy? -62?x]

Answer 1

The divergence of

$\vec{F}(x,y,z)=\left ( P(x,y,z),Q(x,y,z),R(x,y,z) \right )$

div.F(x, y, z) = V.F

дR div.F(x, y, z) = ӘР дх + дQ ду + дz

a)

P = -2y

Q = x

R = 0

$div.\vec{F}(x,y,z)=\frac{\partial (-2y)}{\partial x}+\frac{\partial (x)}{\partial y}+\frac{\partial (0)}{\partial z}$

div.F(x, y, z) = 0+0+0=0

$div.\vec{F}(x,y,z)=0$

b)

P = y²-2x

Q = 5x²y

R = x+3z

$div.\vec{F}(x,y,z)=\frac{\partial (y^{2}-2x)}{\partial x}+\frac{\partial (5x^{2}y)}{\partial y}+\frac{\partial (x+3z)}{\partial z}$

$div.\vec{F}(x,y,z)= -2 + 5x^{2}+3 = 5x^{2}+1$

$div.\vec{F}(x,y,z)= 5x^{2}+1$

c)

P = 3y-2yx

Q = xy²

R = -6z²x

$div.\vec{F}(x,y,z)=\frac{\partial (3y-2yx)}{\partial x}+\frac{\partial (xy^{2})}{\partial y}+\frac{\partial (-6z^{2}x)}{\partial z}$

$div.\vec{F}(x,y,z)= -2y+2xy-12xz$

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