The Laplacian is an operator in vector calculus. It is denoted using a nabla, ∇ 2 {\displaystyle \nabla ^{2}} For some scalar function that takes in n {\displaystyle n} variables, f ( x 1 , x 2 , x 3 , . . . x n ) {\displaystyle f(x_{1},x_{2},x_{3},...x_{n})} , the Laplacian may be defined as:
∇ 2 f = ∑ i = 1 n ∂ 2 f ∂ x i 2 = ∂ 2 f ∂ x 1 2 + ∂ 2 f ∂ x 2 2 + . . . + ∂ 2 f ∂ x n 2 {\displaystyle \nabla ^{2}f=\sum \limits _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}={\frac {\partial ^{2}f}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}f}{\partial x_{2}^{2}}}+...+{\frac {\partial ^{2}f}{\partial x_{n}^{2}}}}
In Cartesian coordinates, the function is f ( x , y , z ) {\displaystyle f(x,y,z)} . The Laplacian is:
∇ 2 f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 {\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}}}}
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