Using spherical coordinates, x = p sin ⁡ ( ϕ ) cos ⁡ ( θ...

Kierra Griffith

Kierra Griffith

Answered

2022-11-28

Using spherical coordinates,
x = p sin ( ϕ ) cos ( θ )
y = p sin ( ϕ ) sin ( θ )
z = p cos ( θ )
Show that ( x 2 + y 2 + z 2 ) 3 / 2 = p 3

Answer & Explanation

polarnost0Yi

polarnost0Yi

Expert

2022-11-29Added 7 answers

LHS = ( x 2 + y 2 + z 2 ) 3 / 2
Put the value of x , y , z
= ( ( p sin ϕ cos θ ) 2 + ( p sin ϕ sin θ ) 2 + ( p cos ϕ ) 2 ) 3 / 2
= [ p 2 ( sin 2 ϕ cos 2 θ + sin 2 ϕ sin 2 θ + cos 2 ϕ ) ] 3 / 2
= ( p 2 ) 3 / 2 ( sin 2 ϕ ( cos 2 θ + sin 2 θ ) + cos 2 ( ϕ ) ) 3 / 2
= p 3 ( sin 2 ϕ + cos 2 ( ϕ ) ) 3 / 2
= p 3 ( 1 ) 3 / 2
= p 3 = R . H . S .

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