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Statements
∇
2
f
=
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
=
1
ρ
2
∂
∂
ρ
(
ρ
2
∂
f
∂
ρ
)
+
1
ρ
2
sin
2
θ
∂
2
f
∂
ϕ
2
+
1
ρ
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
.
superscript
∇
2
𝑓
partial-derivative
𝑓
𝑥
2
partial-derivative
𝑓
𝑦
2
partial-derivative
𝑓
𝑧
2
1
superscript
𝜌
2
partial-derivative
𝜌
superscript
𝜌
2
partial-derivative
𝑓
𝜌
1
superscript
𝜌
2
2
𝜃
partial-derivative
𝑓
italic-ϕ
2
1
superscript
𝜌
2
𝜃
partial-derivative
𝜃
𝜃
partial-derivative
𝑓
𝜃
{\displaystyle{\displaystyle\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2%
}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z%
}^{2}}={\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{%
\partial f}{\partial\rho}\right)+\frac{1}{\rho^{2}{\sin^{2}}\theta}\frac{{%
\partial}^{2}f}{{\partial\phi}^{2}}}+\frac{1}{\rho^{2}\sin\theta}\frac{%
\partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}%
\right).}}
∂
f
∂
x
partial-derivative
𝑓
𝑥
{\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
∂
x
𝑥
{\displaystyle{\displaystyle\partial\NVar{x}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
ϕ
italic-ϕ
{\displaystyle{\displaystyle\phi}}
ρ
𝜌
{\displaystyle{\displaystyle\rho}}
θ
𝜃
{\displaystyle{\displaystyle\theta}}
