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In multiple dimensions, the Laplacian $\Delta u$ is defined as the sum of the unmixed second partial derivatives of the scalar function $u$ with respect to each independent coordinate variable $x_1, x_2, \ldots, x_n$. Mathematically, $\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2}$.

Conditions: $u$ is a twice-differentiable scalar function of $n$ variables.; The coordinates $x_1, \ldots, x_n$ form an orthogonal system (implied by the simple sum).; The definition applies to any dimension $n \ge 1$.

In multiple dimensions, the Laplacian $\Delta u$ is defined as the sum of the unmixed second partial derivatives of the scalar function $u$ with respect to each independent coordinate variable $x_1, x_2, \ldots, x_n$. Mathematically, $\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2}$.

Conditions: $u$ is a twice-differentiable scalar function of $n$ variables.; The coordinates $x_1, \ldots, x_n$ form an orthogonal system (implied by the simple sum).; The definition applies to any dimension $n \ge 1$.

The notation $\nabla^2$ (nabla squared) is mathematically equivalent to $\Delta$ (delta) when representing the Laplacian operator. Both symbols denote the sum of the unmixed second partial derivatives of a scalar field.

Conditions: The operator acts on a scalar field $u$.; The context is vector calculus or PDEs.; $\nabla$ is the del/nabla operator.; $\Delta$ is the Laplace operator.

The notation $\nabla^2$ (nabla squared) is mathematically equivalent to $\Delta$ (delta) when representing the Laplacian operator. Both symbols denote the sum of the unmixed second partial derivatives of a scalar field.

Conditions: The operator acts on a scalar field $u$.; The context is vector calculus or PDEs.; $\nabla$ is the del/nabla operator.; $\Delta$ is the Laplace operator.

The upside-down triangle symbol, known as nabla ($\nabla$), represents the del operator. In the context of Laplace's equation, its square ($\nabla^2$) is equivalent to the Laplacian operator ($\Delta$).

Conditions: The symbol $\nabla$ is the vector differential operator.; $\nabla^2$ denotes the composition of the divergence of the gradient, resulting in a scalar operator.; The equation is the Laplace equation.

The upside-down triangle symbol, known as nabla ($\nabla$), represents the del operator. In the context of Laplace's equation, its square ($\nabla^2$) is equivalent to the Laplacian operator ($\Delta$).

Conditions: The symbol $\nabla$ is the vector differential operator.; $\nabla^2$ denotes the composition of the divergence of the gradient, resulting in a scalar operator.; The equation is the Laplace equation.